exchange (which may be manually or automatically operated) [35]. ..... The continuous baseband signal r(t) is then sampled at the baud rate to g1ve the received ...
Loughborough University Institutional Repository
The adaptive adjustment of digital data receivers using pre-detection filter This item was submitted to Loughborough University's Institutional Repository by the/an author.
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•
A Doctoral Thesis.
Submitted in partial fulfilment of the requirements
for the award of Doctor of Philosophy at Loughborough University.
Metadata Record: https://dspace.lboro.ac.uk/2134/27569 Publisher:
c
S.Y. Ameen
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LOUGHBOROUGH UNIVERSITY OF TECHNOLOGY LIBRARY AUTHOR/FILING TITLE
---------- _.fi':t>€.'f~;---'$-~-------------------------------------------------------------- -----ACCESSION/COPY NO
----------------VOL NO
0~~-2.-t:o&"O:L
--------------------------------CLASS MARK
2
N 1!196
- 1 J UL 1994
3 0 JUN 1995 '
009 4205 02
THE ADAPTIVE ADJUSTMENT OF DIGITAL DATA RECEIVERS USING PRE-DETECTION FILTER
BY SIDDEEQ YOUSIF AMEEN
A Doctoral Thesis submitted in partial fulfilment of the requirements of the award of the degree ofDoctor of Philosophy of the Loughborough University of Technology
May 1990
Supervisor: Dr. S. C. Bateman Department of Electronic and Electrical Engineering
,,
@by S. Y. AMEEN
---~
. --- . "!
'~J~.f·:tA)ii)l!bfll l~ t:li,"';S,i/ ., .)f r.~h.nol-e~\ U:Jr Jry
~- ~~;=
ACKNOWLEDGMENTS I would like to express my thanks to my supervisor Dr. S. C. Bateman for his mvaluable help, advice and encouragement throughout this project without which this work could not have been completed. My director of research, Professor A. P. Clark, was the architect of this research project. Always at hand for advice, conversation and cntical evaluauon of theories and results. For all this and much more, I am deeply grateful and very sorry for his death. The financial support of IRAQI government which made this work poSSible is gratefully acknowledged. On personal level I would hke to thank my family for therr support and encouragement. Most important of all Mona, my wife, who has shown wonderful patience, help, encouragement and understandmg throughout the course of this work.
1
ABSTRACT This thesis is concerned with the adaptive adjustment of digital data receivers employed m synchronous senal data transmission systems that use quadrature amplitude modulation The receiver employs a pre-detection filter that forms the first part of a decision feedback equalizer or else is used ahead of a near maximum-likelihood detector. The filter is Ideally adjusted such that the sampled Impulse response of the channel and filter is mimmum phase. In earlier versions of the filter, when used as part of a decision feedback equalizer it was adjusted by means of the gradient (LMS) algorithm to mimmize the mean-square error m the signal at the detector input. Alternauvely, a more recent technique has enabled the filter to be adJusted directly and in a relatively simple manner from the estimate of the sampled impulse response of the linear baseband channel, which can be determined both accurately and rapidly. The relauve performances and merits of both earlier and recent techniques are investigated for different telephone channels and at transmission rates of9600 and 19200 bit/s. The results confirm that both techniques achieve nearly the same performance at the higher signal-to-noise ratios. The thesis then considers several developments of the more recent technique All of these require a knowledge of the roots of the z-transform of the sampled impulse response of the channel that he outside the unit circle m the z-plane. Several novel algonthms to identify and locate the given roots have been mvestigated Fmally, the technique has been developed for use over HF radio links. Several modified verswns of the original algorithm previously investigated for telephone channels, have also been developed and studied.
11
LIST OF CONTENTS
~
1
LIST OF PRINCIPAL SYMBOLS
vii
INTRODUCTION
1
1.1
BACKGROUND
1
1.2
OUTLINE OF INVESTIGATION
4
2
DATA
TRANSMISSION
SYSTEMS
OVER
THE
7
TELEPHONE NETWORK 2.1
INTRODUCTION
2.2
GENERAL MODEL OF THE DATA TRANSMISSION
7
SYSTEM 2.3
7
ASSESSMENT OF THE DISTORTION PRESENT IN THE SAMPLED IMPULSE RESPONSE OF A BASEBAND
~
CHANNEL
10
2.4
TELEPHONE CIRCUITS
11
2.5
ATTENUATION AND GROUP DELAY DISTORTION
14
2.6
NOISE
15
2.7
MODEL OF THE DATA TRANSMISSION SYSTEM USING QAM SIGNAL
2.8
-
SAMPLED IMPULSE RESPONSES OF CHANNELS USED IN COMPUTER SIMULATION$
~
3
-
CHANNEL EQUALIZATION
3.1 " 3.2
3.3
17
23
40
INTRODUCTION
40
LINEAR EQUALIZER
40
NONLINEAR EQUALIZER
44
lll
34
'
4
DECISION FEEDBACK EQUALIZER
46
3.4.1
Zero Forcing (ZF) Equalizer
48
3.42
Minimum Mean-Square Error (MMSE) Equalizer
51
3.4.3
Comparision of Equalizers
54
a
In the presence of phase distortion
54
b
In the presence of amplitude distortion
57
ADJUSTMENT OF THE PRE-DETECTION FILTER
63
4.1
INTRODUCTION
63
4.2
ADJUSTMENT SCHEME 1
65
43
ADJUSTMENT SCHEME 2
70
4.4
ADJUSTMENT SCHEME 3
73
4.5
ADJUSTMENT SCHEME 4
74
4.6
ASSESSEMENT OF RESULTS AND DISCUSSION
76
5
ALGORITHMS FOR Tf-JE ADJUSTMENT OF THE PRE-DETECTION FILTER
96
5.1
INTRODUCTION
96
5.2
ROOT-IDENTIFICATION ALGORITHMS
98
5.2.1
Schur Algorithm
98
5.2.2
Nyquist Criterion
101
5.3
ROOT-FINDING ALGORITHMS
105
5.3 1
Algorithm 1
105
5.3.2
Algorithm2
111
5.3.3
Algonthm 3
113
5.3.4
Algonthm4
117
5.3.5
Algorithm 5
118
IV
5.4
' 6
ASSESSMENT OF RESULTS AND DISCUSSION
ADAPTIVE DECISION FEEDBACK EQUALIZERS
121 159
6.1
INTRODUCTION
159
6.2
FEEDFORWARD TRANSVERSAL FILTER CHANNEL ESTIMATOR
159
ADAPTIVE DECISION FEEDBACK EQUALIZERS
163
'63 63 1
Equalizer 1
163
6.3.2
Equalizer 2
167
6.3.3
Equalizer 3
170
64 7
ASSESSMENT OF RESULTS AND DISCUSSION
ADAPTIVE ADJUSTMENT OF THE RECEIVER OVER HF RADIO LINKS
172
189
7.1
INTRODUCTION
189
7.2
MODEL OF SYSTEM
189
7.3
ADJUSTMENT ALGORITHMS
194
7.4
7.5
7.3.1
Adjustment algorithm 1
196
7.3.2
Adjustment algorithm 2
197
7.3.3
Adjustment algorithm 3·
198
7.3.4
Adjustment algorithm 4
199
7.3.5
Adjustment algorithm 5
201
DECISION FEEDBACK EQUALIZERS OPERATING OVER HF CHANNELS
204
ASSESSMENT OF THE RESULTS AND DISCUSSION
207
V
......
8
'
CONCLUSION
230
8.1
COMMENT ON THE RESEARCH
230
8.2
SUGGESTION FOR FURTHER INVESTIGATION
232
REFERENCES
234
APPENDICIES LIST OF COMPUTER SIMULATION PROGRAMS
242
APPENDIX 1 CALCULATION OF THE SAMPLED IMPULSE RESPONSE
OF
THE
LINEAR
BASEBAND
'
CHANNEL OVER TELEPHONE NETWORKS
242
APPENDIX 2 LINEAR PRE-DETECTION FILTER ADJUSTMENT SCHEME 1
249
APPENDIX 3 LINEAR PRE-DETECTION FILTER ADJUSTMENT SCHEME2 APPENDIX 4
253
LINEAR PRE-DETECTION FILTER ADJUSTMENT SCHEME4
APPENDIX 5
THE
256
SCHUR
ROOT-IDENTIFICATION
ALGORITHM APPENDIX 6 THE
APPENDIX 7
265
NYQUIST
ROOT-IDENTIFICATION
ALGORITHM
268
ROOT-FINDING ALGORITHM 4
270
APPENDIX 8 DECISION FEEDBACK EQUALIZER 1
277
APPENDIX 9 ADJUSTMENT ALGORITHM 4 OVER HF RADIO LINKS
288
APPENDIX 10 DECISION FEEDBACK EQUALIZER H5 OVER HF RADIO LINKS
297
VI
LIST OF PRINCIPAL SYMBOLS a(t) & A(f)
Impulse response and transfer function of a filter. Attenuation of telephone circmt and eqmpment filter at frequency f. z-transform of one-tap feedback transversal filter in Fig. 5.22.
B
(n+g+ I)-component row vector given by Eqn. 3.45. z-transform of two-tap feedforward filter in Fig 5.40.
b(t) & B(f)
Impulse response and transfer function of b filter.
c
Positive constant.
c
Equalizer tap gains in Sections 3.1 and 6 3.3.
c(t) & C(f)
Impulse response and transfer function of bandpass filter C.
d
Threshold level for determmmg convergence of iterative process.
D(z)
z-transform of hnear pre-detecnon filter.
D'(z)
z-transform of reversed order sequence corresponding to D(z).
E[x]
Expected value of x.
E(z)
z-transform of the combined sampled impulse response of the channel and pre-detection filter.
e,
Error signal at t=iT. Carrier frequency m QAM system in Hertz.
/(z)
First denvative of f(z).
((z)
Second derivative of f(z).
g+l
Number of components in a sampled impulse response of the hnear baseband channel.
g(t) & G(f)
Impulse response and transfer function of the filter G.
gd,(f)&gd,(f)
Group delay of telephone circuit and equipment filter.
gd
Average group delay.
Vll
h(t) & H(f)
Impulse response and transfer function of the telephone circuit.
CL(j), HL(j) & GL(j) Low frequency components of C(f), H(f) & G(f).
n+l
Number of taps of the lmear pre-detectiOn filter.
n(t)
White Gaussian noise waveform at the input of telephone circuit.
r(t)
Received waveform.
{r,}
Sequence of received samples that is obtruned by samplmg r(t) at t=iT, i=l, 2, ..... .
R,(f)
Transfer function of the receiver filter set at frequency f given by Eqn. 2.20.
f,
Estimate of r,.
s,
The 1m transmitted data symbol.
s,
The detected value of s,.
Sa,l'
sb,•
The real and the imaginary part of s,
s,
(n+g+ I)-component row vector given by Eqn. 3 37.
T,(f)
Transfer function of the transmitter filter set given by Eqn. 2 19.
Um(z)
Deflated form of the polynorrual Y(z) after removal of m roots.
{u.}
Noise samples at the pre-detection filter output.
v,
Output signal from the linear pre-detection filter.
w(t)
A Gaussian random process with zero mean.
{w,}
A sequence obtained by sampling w(t) at t=iT, i=l, 2, ....
x(t)
QAM signal given by Eqn. 2.6.
y
Sampled impulse response of the linear base band channel.
Y(z)
z-transform of the sampled Impulse response of the lmear base band channel.
y
Estimate of Y.
Yr(z)
Truncated form of Y(z).
Vlll
Factor of Y(z) with all its roots inside the unit circle. Factor of Y(z) with all its roots outside the unit circle.
z.
(n+ 1)-upper triangular square matnx given by Eqns. 3.46 and 4.38.
o(r)
Dirac function (unl!Impulse). Variance of u,.
8
posotlve constant. Mean-square value of the real or Imaginary part of s,. Variance of real or imaginary part of w,. Parameter given in Eqns. 3.49, 4.21 and 7.24. Mean-square error due to additive nmse only given by Eqn. 4 20. Parameter given in Eqn. 6.12. Parameter giVen in Eqns. 4.18 and 6.1 0. Parameter given in Eqns. 4.19 and 6 11. Parameter given m Eqns. 7.18.
e.
Parameter given in Eqn. 5.22. Negative of the roots of Y(z).
~)
Negative of the reciprocal of a root of Y(z) that he outside the umt crrcle.
p(t)
Stationary zero-mean real-valued Gaussian random process. Estimate of the negative of the reciprocal of the m'h roots at the 1terat10n.
i''
Estimate for the biggest roots of Yr(z) Threshold level for determimng convergence of iterative process. Estimate of ~•.
ix
Phase angle. Channel estimator step s1ze (Eqn. 6.3). Variation around the contour C. Sampling penod in frequency domam. Gradient algonthm step size.
e
Variation of around the unit circle Transmitted data. Detected value of Xk·
*
ConvolutiOn operator.
super script *
Complex conjugate.
super script T
Transpose.
X
CHAPTER!
INTRODUCTION
1.1 BACKGROUND
The widespread use and the increasing availabllity of low-cost digital data transmission systems has created an mcreasing demand for more efficient methods of transmissiOn. Furthermore, the rapid advances in digital computers and silicon technology have increased the reqmrement fortransmittmg data signals at the highest rates over a wide range of communication channels This in turn has made the problem of digital communication through hnear channels that exhibit both amplitude and phase distortion important and practical. Different media are used for digital data transmission, but the most Important of these due to therr very widespread existence are voice-frequency channels over the telephone network and HF radio links [1-10]. The public switched telephone network (PSTN) was used pnmanly to carry analogue speech signals only [3-5,8]. It followed that the spectral characterisncs of such channels were optimized, over time, to achieve a good noise performance and an acceptable quality of service, the latter being a highly subjective measure [3,5]. However, the use of these voice-frequency channels for transmission of greater volumes ofhigher-speed data traffic has shown that very specialist and sophisticated methods of signal processing must be employed at the receiver m order to achieve an acceptable error rate performance, which IS, itself, a highly definable measure [3,9,10] In essence, amplitude and phase distortion present m a data signal will cause It to spread out in time, eventually resulting in intersymbol interference, where adJacent data elements overlap in time [3,10-18]. It is m this sense (1 e taking into account the distortion and the recognition that different channels will introduce different amounts of distortiOn), that modern, high-speed digital data receivers must be adaptive m structure [3,13-18]
1
•
The high-frequency (HF) (nominally 3-30 MHz) portion of the spectrum has been of great interest for long distance radio commumcations in a variety of nulitary and CIVIlian applicatiOns for many years [3,5,7,14,17-24]. These have different properties from telephone circuits and they are normally used for point-to-pomt communications, or for isolated communication networks, and not often used as a part of the general telephone network [3,5,17]. However, HF radio lmks are subject to ionosphenc multipath propagation and fading, which makes the problem of communication over these channels difficult, even at moderate rates [5,7] The major limitations for high data rate HF transmissiOn are a result of the non-Ideal characteristics of the medmm, such as lmear distortion, rapid channel variations and severe fadmg as well as bandwidth constraints [3,5,7,25]. Therefore, m high speed digital commumcatwns, the efficient use of the available bandwidth IS often linuted by the presence of intersymbol interference caused by the non-Ideal channel charactenstics. Clearly, high speed data transmission can be severely degraded by the presence of intersymbol mterference unless the receiver employs complex equalization techmques and multi-level signallmg [3, 11, 14, 16,27] The lattens achieved by using an efficient modulation technique such as quadrature amplitude modulation (QAM), which has been used since the late 1960's to achieve reliable transmission of data at rates of upto 9600 bit/s and more [26]. In practice, and due to their easy implementatiOn using cheap, medmm-technology components, linear equalizers are often used [27-29]. For channels introducing a severe degree of bandlimiting, coupled with the requrrement of a high transmission rate, a sigmficantly better performance can be achieved under comparable operation conditions, by the use of a pure nonlinear equalizer [27 -30]. The latter uses a method of decisiOn directed cancellation of mtersymbol interference and can suffer from error extension effects [27]. An alternative method of equalization is to use the deCision feedback equalizer. This type of equalizer mcludes an adaptive lmear pre-detection filter placed ahead of the nonlinear equalizer structure itself. The decision feedback equalizer often achieves a better tolerance to nOise than a lmear and a pure nonlinear equalizer [3,27]. However, computer simulations studies have shown that partial equalization of the channel, along With the use of a fmrly sophisticated detection scheme such as near maximum-likelihood detection may
2
well result in a system which is cost effective (in terms of computational effort) and also gives a supenor performance when compared with classical equahzanon techniques [ 11,17 ,31]. The decision feedback equalizer and the partial equalization that is employed by the near maximum-likelihood detector both use a lmear transversal filter called the pre-detection filter. In the earher versiOns of the filter, it was adJusted by means of the gradtent (least mean-square error) algorithm or equivalent techniques to mmimize the mean-square error m the output stgnal at the detector mput [3,27,29,31-33]. Unfortunately, when the channel introduces severe amplitude dtstortion, correct convergence is not necessanly achieved by this arrangement, and the rate of convergence may be very slow [3,32]. Altemauvely, undue complexity may be involved in the system [17] However, a recent development has enabled the filter to be adjusted drrectly and in a relatively simple manner from the esumate of the sampled impulse response of the channel, which can be determined both accurately and rapidly even in the presence of severe amplitude distortion [3,17,33-38]. Thts has opened the way to widespread research. It is essentially a root-findmg algonthm that determines, m sequence, the roots (zeros) of the z-transform of the sampled impulse response of the linear baseband channel, that lie outside the umt ctrcle in the z-plane It then uses the knowledge of these roots to determine the tap gains of the linear pre-detection ftlter and to form an estimate of the sampled impulse response of the channel and filter. The pre-detection filter at all times introduces an orthogonal transformanon into the received signal, bemg constramed to be an allpass network [17,27 ,39]. When the filtens correctly adjusted, the zeros of the z-transform of the sampled impulse response of the channel and filter are denved from those of the z-transform of the sampled impulse response of the channel by replacmg all zeros of the latter that he outside the umt circle by the complex conjugate of thetr rectprocals, all remaimng zeros bemg left unchanged. The mm of the work presented m this thesis is to develop satisfactory methods that can be used m the adJUStment ofthe pre-detection ftlter for use over telephone circuits and HF radiO links which are being used for transmission of data at high speeds (9600 bit/r and above over voice-frequency channels). The study also aims to compare the performance of the developed methods with more conventional soluuons.
3
1.2 OUTLINE OF INVESTIGATION
Essentially, the investigation is concerned with the senal transmission of data at 9600 bJt/s over telephone crrcmts and HF radio links. Over telephone crrcuits, a feas1billty study has been earned out in this work, looking mto the poSSibility of transmission at rates close to the Shannon limit, i.e. 19200 bit/s [40]. The main concern has been the development of methods for the adJustment of the pre-detectwn filter that can be employed by the decision feedback equal1zer or near maximum-likelihood detector. The research has been earned out by computer simulation of the modem. In th1s situatiOn, computer simulatiOn IS a valid means of evaluating the system performance because the modem can be considered as a digllal s1gnal processor performing computer like operations on set of numbers Chapter 2 provides a detailed description of the general model of a synchronous senal data transmission system. The nature of telephone Circuits and the types of signal distortion and other impairments introduced by these circuits along w1th the types and effects of noise introduced are also presented and d1scussed. The suitability of quadrature amplitude modulation (QAM) is discussed and the system model using 16-level and 64-level QAM signal constellations are also d1scussed. The model of the system using QAM signals is then presented together wJth the relauonsh1ps between the bandpass processes and signals, wllh their equivalent baseband representauon. Fmally, the sampled impulse responses of the equivalent lmear base band channels when the data symbols transmllted at 2400 and 3200 baud over d1fferent telephone circuits are calculated and recorded. Chapter 3 contains the descriptions of the linear, pure nonlinear and decision feedback equalizers These equalizers employ linear feedforward transversal filters and can be used as techniques for processing distorted received s1gnals The decisiOn feedback equalizer employing the pre-detection f1lter achieves better performances than both linear and pure nonlinear equalizers and can be adJusted eJther to minimize the mean-square error in the equalized signal or max1mize the signal-to-nmse ratio at the detector input. The latter is achieved subject to the exact equalizauon of the channel. Theoreucal comparison between the two type of decision feedback equalizers under d1fferent circumstances are then presented.
4
Chapter4 investigates several schemes that adaptively adJUSt the pre-detection filter. The filter is adJusted using two criteria, the first maximizes the signal-to-noise ratio at the input of the detector subject to exact equalization of the channel and the second mmimizes the mean-square error in the signal at the detector input. Three adjustment schemes are presented which adjust the filter according to the former cntena and a fourth scheme adjusts the filter according to the minimum mean-square error cnteria. The study aims to show the relationship between both cnteria and the complexity of the firSt three schemes. Computer Simulation studies when operating over different telephone channels are presented for various adjustment schemes, and their performances, in terms of equalization accuracy and complexity, are investigated. Chamer 5 descnbes different algorithms that have been used in the adjustment mechanism of the pre-detection filter to locate or identify all or some of the roots of the z-transform of the sampled impulse response of the linear baseband channel. These algorithms operate directly on an estimate of the raw sampled Impulse response of the channel and can, by design, be made to operate outside the normal flow of processing steps reqmred in the receiver of a data modem. The estunate of the raw sampled Impulse response is taken to be ideal estimate. For the requirement of the digital receiver to the estimate of the intersymbol interference, some of these algorithms have an extra stage, where the estimate of the sampled Impulse response of the channel and pre-detection filter is calculated. Results of computer simulation tests are then presented showing the convergence rate and the accuracy of the iterative processes. The number of numerical operations involved in the execution of each algorithm are also shown. The latter measure IS extremely Important
when
considering the development of the modem using practical digital signal processmg devices. Chapter 6 first discusses the adaptive estimation of the sampled impulse response of the linear baseband channel. It then extends the study to mclude the adaptive adJustment of the decision feedback equalizer. The decisiOn feedback equahzer here is adjusted to maximize the signal-to-noise ratio at the detector mput subject to the exact equahzation of the channel. Results of computer simulation tests over different models of telephone channels are then presented showing the channel estimator performance. The effect, of the channel estimator together with the finite number of taps in the pre-detection filter, on the equalizer performance were also
5
--------------------------------------------------------
--
--
-
invesngated. Fmally, the tolerance of the equahzerto additive white Gaussian noise was computed and compared with the Ideal case and with the cases where the equalizer was adjusted by the gradtent (LMS) or the Kalman (RLS) algorithms. In the ideal case, perfect estimanon of the sampled impulse response of the channel, theoretical root-finding algorithm and mfmite number of taps in the pre-detection filter were assumed. Chapter 7 first presents the model of the data transmission over HF rad10 lmks . Several modified verswns of the origmal algonthm previously invesngated for telephone channels have been developed and studied for use over HF radiO lmks. Results of computer simulation tests showmg the accuracy and the complexity of the algorithms over HF radiO links, that introduce vanous levels of distortion, are calculated and presented. Finally, the effects of these algorithms on the tolerance of the decision feedback equalizer to additive white Gaussian noise, when the algorithms are used to adJUSt the equalizer, were also investigated for different HF radio links
6
CHAPTER2
DATA TRANSMISSION SYSTEMS OVER THE TELEPHONE NETWORK
2.1 INTRODUCTION
In the last three decades the demand for efficient and high speed data transmissiOn systems has mcreased. Comprehensive studies to put mathematical models of reliable data transmission systems have been based on Shannon's result on the maximum transmission limits over the channel [40]. This chapter presents the basic elements and provides a brief introduction to the general requirements of a digital data transmissiOn system operating over the telephone network. Particular emphasis is given to the derivation of the baseband model of QAM systems, employmg linear coherent demodulation at the receiver. This model reduces the cost of buildmg and testing the hardware of the new or improved digital commumcation system by allowing the system to be simulated using a digital computer. 2.2 GENERAL MODEL OF THE DATA TRANSMISSION SYSTE:M The system model under investigation
IS
as shown in Fig. 2.1. The data
communication system may be a serial or parallel system [5]. A serial system is one in which the transmitted signal compnses a sequential stream of data elements whose frequency spectrum occupies the whole of the available bandwidth of the transmission path. A parallel system IS one m which two or more sequential streams of data signals are transmitted simultaneously, and the spectrum of an md1vidual data stream normally occupies only a part of the available bandwidth [3,5,16,27] In a serial system the signal elements are normally transmitted at steady rate of a giVen number of element per second (baud). The receiver extracts the element tumng mformation from the received signal and operates m time synchronism with the received signal. Such a system is called a synchronous serial system [5]. A serial data transmission system is less complex than a parallel data transmissiOn system
7
as the latter needs several demodulators to process the different signals [5,12]. In application where a relatively high transmission rate is reqmred over a given channel, a synchronous serial system is the most commonly used system [5,12]. Therefore, it will be assumed throughout this work that the data transmission system is a synchronous senal data transmission system. A sequence of regularly spaced weighted impulses I.s,o(t -1T) representing the
' transmitted data signal elements are fed to the input of the transmitter filter with a baud rate of 1(f. O(t) is the dirac function and S, IS the
lth
transmitted data symbol
which may have m possible values;
s, = 21-m+1
1=0,1,
.
,m-1
21
The {s,} are considered to be statistically independent symbols and are assumed to be equally hkely to take on any one of m possible values in any one symbol period. The transmitter filter limits the spectrum of the transmitted signal energy to the approximate available bandWidth of the transmission path. The receiver filter removes the nOise components outside the frequency bandwidth which approximately corresponds to the bandwidth of the received signal [5,12,35]. The transmission path could be either a lowpass channel, with a frequency lirrut no greater than 10 kHz or a typical voice-frequency channel with a frequency band no wider than about 3 kHz such as could be obtained over the telephone network or an HF radio hnk [3,5,12,16,27]. In the latter-case, the transmission path in Fig. 2.1 is assumed to include a lmear modulator at the transrrutter and a hnear demodulator at the receiver. The transmitter filter, the transmission path and the receiver filter m cascade are assumed here to form a lmear baseband channel whose Impulse response IS y(t); for practical purposes, y(t) is assumed to be of finite duration and time mvariant over the mterval of any transmission. The only nOise mtroduced by the channel Is addltlve noise which can, for practical purposes, be taken to be additive white Gaussian nOise [3,5]. It has been shown that If one data transmission system has a better tolerance to additive white Gaussian
8
-----------------------------------
-
noise than another, it will also, in general, have a better tolerance to other type of additive noise introduced over telephone circuits [3,5, 12-13]. Furthermore, the effects of addmve white Gauss1an nmse on a digital data transmisswn system may readily be analysed theoretlcally and stud1ed by computer simulation. For these reasons, it will assumed that the only noise introduced at the output of the transmission path in the model of the data transmission system is additive white Gaussian noise. The nmse has a zero mean value and a two-sided power spectral density of ~N0 , giving the zero mean Gauss1an waveform w(t) at the output of the receiver [5]. Thus the output signal from the baseband channel in Fig 2.1 is the waveform [17]
=
r(t)
L:s,y(t- iT)
+ w(t)
2.2
where {s.} are the values of the transmllted data elements, y(t) is the impulse response of the equivalent base band channel, and T 1s the symbol duratwn. The continuous signal, r(t), at the output of the baseband channel is sampled, at the baud rate, to give the received samples {r,}, where 1 takes all possible integer values. Various techniques for holding the samplmg instant correctly synchronised to the received signal are given elsewhere [41-42]. The i'• rece1ved sample is therefore; g
r,
where r,
= h=O 2: s,_.y. =r(zT),
Y•
+ w,
2.3
=y (hT), w, =w(zT) and Y•
is the (h
+ 1)1• component of the
sampled 1mpulse response of the baseband channel. The delay in transrmsswn, other than that mvolved m the time d1spersion of the transmllted signal, is neglected here, so that y 0 "' 0 and Y• = 0 for hg. The sampled impulse response of the lmear baseband channel, or at least an accurate estimate of lt, 1s assumed to be known at the receiver and 1s given by the (g+ I)-component row vector, Y, which is g1ven by,
Y
=
[
Yo Yt
· Y,
9
]
2.4
In the signal processor and detector m Fig. 2 1, the values of the {.f,} are determined from the {r.} using any one of several different detection and equalization techniques. The signal processor IS assumed here to have prior knowledge of both Y and the possible values of the s,. The detected data value of s, IS designated to be §,
2.3 ASSESSMENT OF THE DISTORTION PRESENT IN THE SA:\IPLED IMPULSE RESPONSE OF A BASEBAND CHANNEL Two types of distortion can be present in a sampled waveform; these are amplitude and phase distortion [3,5,10-13,27]. For no amplitude distortion, all the amplitudes of samples at each frequency are scaled by a constant amount [5,10-12]. Therefore, the attenuation is independent of frequency. For no phase distorllon, the wave velocity must be linear [1 0-12]. Therefore, a graph of phase against frequency must be a straight line. Group delay is a common measure related to the phase distortion [5,12]. It is defined as the rate of change phase with frequency [5,10-12] Phase d1stort10n can be defined in terms of the components of the DFT (D1screte Fourier Transform) of the sampled impulse response of the correspondmg channel. When all the components of the DFT of the channel vector Y have all the magnitudes equal to unity, the channel represents pure phase distortion [10-12]. The deviation of these magnitudes from unity can be used to esnmate the level of amplitude distortion present Alternanvely, the aperiod1c autocorrelation functiOn has all components equal to zero except for the (g + 1)'• component, which has the value of unity [10-12]. These properties sho_w that pure phase distortion represents an orthogonal transformation [12,27]. Therefore, a suitable detecnon process provided at the receiver can reverse the orthogonal transformanon, phase distortion is not usually considered when assessing the distortion present in the sampled impulse response of the channel [27]. Amplitude distortiOn is a much more serious consideration because it always lowers the tolerance to noise of a data transmission system [12,27]. In terms of the components of the DFT of Y, the magmtude of the components are not equal to umty. Another important feature of amplitude distortion
IS
that it always changes
the discrete energy-density spectrum and the apenod1c autocorrelanon function of
10
the sampled impulse response of the channel [12,27]. The d1screte energy spectrum IS
the DFr of the a periodic autocorrelation function [10-12]. Pure amplitude
distortion means no phase distortion or delay. In order for the latter condmon to be sat1sfied, the components of DFr must be all real-valued [12,27]. These properties show that pure amplitude distortion is a symmetnc transformation [10-12,27]. In terms of the zeros and poles of z-transform of the sampled impulse response of the channel, ampl!tude and phase distortion can also be defined. When the channel introduces pure phase distortion, each zero and pole of the z-transforrn of Y are accompanied by a pole or zero, respectively, at the complex conJugate of the rec1procals value of z [27]. Whereas, when the channel introduces pure amplitude distortion means that the z-transform of Y contains an even number of zeros, each zero being accompanied by another zero at the complex conjugate of the reciprocal value of z [27].
2.4 TELEPHONE CIRCUITS Telephone circuits are an arrangement w!lh electrical interconnecting whereby the communicatiOn of speech or data can be carried between any two points. It may be pnvately or publicly owned. Pnvate circmts can be considered as point to point communical!on wh1ch are permanently or on a part time bas1s rented by one or more subscribers [3-6,43-46]. They are not connected through any of the switches in the exchange nor to the exchange or repeater stauon's battery supplies. Large organisal!ons such as banks and rarlway authonties may have their own network of l!nes to meet therr own demands [43] The publ!c sw!lched telephone network (PSTN) 1s necessary to connect any given subscriber to another at a telephone exchange (which may be manually or automatically operated) [35]. These services are most useful where the transmission time per day Will be relatively short, or when a central point has to communicate w!lh a large number of outstations [46]. The chmce between using private circuits or using the PSTN for data transffilssion must be made by careful consideration of factors, such as the costs, availability, speed of workmg and transmission performance. A PSTN connection
IS
establ!shed by
dmllmg the telephone number of the distant data terminal and the route used for a particular call1s a random chmce from a large number of d1fferent poss1ble routing,
11
includmg various combinations of audio-frequency cables and mulu-channel systems [45-46]. For data transmission, private-leased crrcuits have the following advantages over the PSTN [3-6,35,43-46]. The bandwidth ofpnvate circmts tends to be around 300-3000 Hz, v.h1ch can
i-
be adjusted to give a good performance over this bandwidth, whereas the bandwidth of the PSTN
IS
more restncted, nominally 300-500 Hz and
900-1200 Hz, to avmd the frequenc1es used for in-band signalling on truck routes n-
private circmts are less nmsy than PSTN channels because of the switching equipment used in telephone exchanges.
m-
Eff1cient performance due to exclus1ve use of the Circuit is obtained.
iv-
The link can be adjusted to have the optimum performance, where high speed and more reliable transmission are possible.
v-
Full-duplex operauon IS available at higher bit rates
Fmally, the cost of the permanently private circuit may be relatively high especially if there are insufficient data traffic on the line. In practice, most private data networks consist of a combinations of both leased and PSTN lmk. Vmce-frequency data circmts can also be of any length. Unloaded audio lmks are generally very short (of the order of 3 miles) [5]. They comprise a pair of balanced twisted wires with nominal impedance of 600 ohm. They have a good frequency response, w!!h some attenuation distortion and negligible delay distortion over the voice-frequency band [5]. Furthermore, and over a vo1ce-frequency as the length of the link m creases the attenuation increases which prevents the use of long unloaded audio lmks, where the mcrease in attenuation is about 2 dB per miles [5]. In many situations, It is deSirable to extend the length of the links beyond this limit of 3 miles. Common methods to attam longer links without exceedmg loss limits are the followmg [5,43-46];
12
i- Increase the conductor diameter. ii- Use amplifiers. hi- Use inductive loading. Loading a particular voice pair links consists of mserting series inductance (often 44 or 88 mH) into the lmks at a fixed mtervals (typically 2000 yds). Adding load coils tends to; i- decrease the velocity of propagation. Ii- increase the impedance. Over the centre of the vmce-frequency band the attenuatiOn distortion decreases due to the presence of the loading coils; in return the attenuation and group delay distortion will mcrease over the higher frequencies [5,43-46]. This mcreases as the length of the loaded audio link increases, especially at the htgh frequency end of the voiceband [5]. The third type of link is the carrier link which can be very much longer than loaded audio links. The modulation process at the transmitter is a single stde band suppressed-carrier amplitude modulation which shift the stgnal frequency band to some higher frequencies [5]. Because of the high cost of !me plant it is desirable to use a line carry more than one data lmk (multi-channel) by usmg a multiplexer. With frequency divisiOn multiplexing, each data channel is shifted or frequency translated to a different part of the available frequency spectrum. The particular frequency to whtch the channel is shtfted IS determmed by the frequency of the carrier whtch is modulated by the data signal. This combination of modulation and frequency division multiplexmg is the basts of the multi-channel carrier telephony systems which operate over carrier pairs, coaxial cable, microwave radiO relay systems, satelhte systems and HF radio links The distortions introduced m earner ctrcmts are entirely determined by the filters involved in the modulatiOn, demodulation and multiplexmg processes which originate at the termmal station and do not depend on the length of the lmk [5,10,35]. The correct operation of a multt-channel system rehes upon a earner bemg re-inserted at the receiver at the correct frequency.
13
Although elaborate synchronization circmts are used, the frequency offset may be as much as± 2Hz [5,46]. This leads to signal impairments rather than pure signal distortion.
2.5 ATTENUATION AND GROUP DELAY DISTORTION Signal distortion is the change in shape of the transmitted Signal, resulting from attenuation and group delay characteristics. These two charactenstics are the most widely studied characteristics of telephone connections and can be defined as the variation of the attenuation and group delay with frequency [5,10-12,35]. The attenuation distortiOn in a given frequency band is the variation of attenuation over that frequency band and group delay d1storuon !S the vanauon of group delay over the frequency band A switched line is made up of a number of separate lmks, each selected on a purely random basis from a large number of poss1ble routings [5]. It 1s therefore poss1ble to have echoes or reflected signals, if there 1s any mismatch m the system. This mismatch occurs because it is not possible to check and correct the frequency characterisucs of the complete circuit [5,27]. Such echoes can be extremely annoying and will worsen with mcreased delay. When there are two or more m1smatches along the !me, the received signal element comprises the mam components of the element followed by several echoes, which are attenuated and delayed with respect to the main component. Th1s effect of m1smatch is an example of multipath propagauon, where the received signal arrives at the receivmg end over more than one path. Echoes in the received signal mdicate the presence of the appropnate combination of attenuauon and group delay distortion, where the signal distortion 1s dependent upon the relauve magnitudes of echoes and main Signal components. Echoes can occur on bo.th sw1tched and pnvate circuits although because of the possibility of !me mismatch 1s greater for connecuons over the switched network, severe echoes are more likely to be encounted over the PSTN. F1g. 2.2 shows the attenuation and group delay charactensucs of an ideal voice-frequency channel, where the attenuauon and group delay nse rapidly at frequencies below 300 Hz and above 3400 Hz [5]. Channels introducmg severe attenuation and group delay distortion may have sharper responses [5].
14
-------------------------------------------------------------------------------------1
The attenuation and group delay characteristics causes time dispersion of the received signal which usually mcreases with the attenuation and group delay distortion in the signal frequency band [5,10-12,27,35] The effect of this time dispersion IS to spread out, in time, the response of a data pulse such that It overlaps the adjacent pulses in a digital wave train This IS called the intersymbol interference and reduces the margm against noise or, m severe cases, causes systematic errors [27]. The major effect of time dispersion is to set up an upper limit on the rate at which the element signal transmitted for an acceptable level of mtersymbol interference [35]. Therefore, channels with low levels of attenuation and group delay distortion will be able to transmit the data element at higher rates than those exhibltmg higher attenuation and group delay distortions. As a result, the impulse response of the channel (the signal at the channel output when the mput signal is the dirac function) IS a contmuous rounded waveform of duration not less than fraction of milli-second [10]. When the input signal to the channel is a sequence of impulses {o(t -zT)}, where i=1,2, ... , and T is the time interval separatmg two successive impulses, the output signal IS a sequence of continuous rounded waveforms, each being given by an appropriately delayed version of the channel impulse response. When the timeT is shorter than the duration of the channel impulse response, then consecutive output rounded pulses overlap producing what IS known as intersymbol interference [5]. In practice, a good channel will have an impulse response which a rapid rise to Its peak value followed by a rapid decay [5,10-12,27 ,35]. 2.6NOISE The term noise is used to designate i.mwanted signals that tend to disturb the transmissiOn and processmg of signals in commumcation systems. Telephone circmts mtroduces different types of additive and multiplicative noise [5,47]. Impulsive noise appears in the public switched telephone network as predommant additive noise because of the electncal/mechamcal switches in the exchanges [3-5,10-12,43-46]. Unfortunately, the shape of the impulsive noise varies widely from one telephone circuit to another, with durations extended over many adJacent signal elements, makmg the simulation of this noise by computer a difficult task [5,43-46]. Recently, electncal/mechamcalswitches have been replaced by electronic switches, which greatly reduce the impulsive nmse. Speech and signallmg tone
15
cross-talk is another type of addilive noise wh1ch occur because of capacitance unbalance between the pairs of wires used in a connection [5,43-46]. They are not normally important in causmg errors except at low signal levels or under !me fault conditiOns [5,46]. Wh1te noise, whose spectral density function is flat out to frequenc1es well beyond those occupied by any message bearing s1gnals under consideration, produces errors only at very low Signal level and is not normally a s1gmficant source of error [5,47]. The best examples are thermal nmse and shot noise which have a Gaussian amplitude distributions and are known as Gauss1an wh1te noise [47]. Passing this noise through bandlimiled telephone circuits w1ll produce band-limited while noise or coloured noise [47]. Strictly speakmg, white noise has an inf1mte average power and, as such, it
IS
not phys1cally realizable [47-49].
Nevertheless white nmse has convenient mathematical properties and therefore is useful in system analysis [3,5,10,35,47-49]. Any two different samples of while nmse are uncorrelated [5,12-13,16,47-49]. These samples are statistically independent if the white noise
IS
Gaussian d1stnbuted, because Gaussian nmse
represents the ultimate m randonmess [47-49]. The multiplicative noise involves both amplitude and frequency modulation effects [5,16]. The noise waveform amplitude or frequency modulate the signal waveform. The noise originate on earner lmks
and can be of several types as follows
[3,5,16,43-46]; i- Modulation noise. ii- Transient interruptions. iii- Sudden level changes. iv- Frequency offsets. v- Sudden phase changes. v1- Phase Jillers. Fmally, the majority of noise introduced over the switched telephone network 1s additive. Multiplication noise can be the predominant type of noise introduced over private lines [5].
16
-----------------------------------------------------------------------------
2.7 MODEL OF THE DATA TRANSMISSION SYSTEM USING QAMSIGNAL The model of the digital data transmission system usmg QAM (Quadrature Amplitude Modulation) signals is Illustrated in Fig. 2.3. The QAM system involves the transmission of two parallel signals each requmng a separate amplitude modulator at the transmitter and a separate amplitude demodulator at the receiver [11,27-29]. The two modulated signals have the same carrier frequency but are in phase quadrature. A QAM system is used here because it has several important advantages over other modulatiOn techniques, these are as follows, [10-12,26,35] 1-
Maximum avmlable tolerance to nOise achieved when used wah near optimum detection process.
ii-
Full use of the available frequency band, since the carrier frequency IS at the centre of the available frequency band.
iii-
S1mple equipment filters, since the amplitude and phase characteristics of the filter do not have to have any particular shape to satisfy any very stringent conditions, which is the case in the S S.B (Single sideband modulation), V.S B. (Vestigial sideband modulation) and I.S.S.B. (Independent smgle side band modulation) systems.
1v-
No p1lot earner needs to be transmitted with a QAM signal, since the rate of change of the relative phase angle is farrly small. This reduces the complexl!y of receiver by avoiding the ISohition of the p1lot earner from the data s1gnal at the receiver
The information to be transmitted IS a sequence of binary digits {Xk}, where Xk takes any of Its two poSSible values, 0 or 1. These binary d1g1ts are coded into two multi-level signal s•.• o(t -1T) and sb,,6(t- iT) The real and 1magmary parts of the correspondmg s, are statistically mdependent and equally likely to have any of their m poSSible values, where,
17
s•.• ,
sb,,
= 2/
-
m
+ 1
l
=
0,1,2,
.. ,
m-1. ..
2.5
so that any two data symbol (s.,.,sb,.) may have one of m 2 possible combmations. It has been assumed that, when the bmary digits are fed to the encoder at a rate of 9600 bit/s, the possible values of s•.• and sb,• are± 1 and± 3. It is also assumed that, when the bit rate is 19200 bit/s, the possible values of s••• and sb,, are± 1, ± 3, ± 5 and±7. Each of the two data streams s•.• and sb,• are fed separately to one of the lowpass filters at the transmitter, to be shaped to the appropnate bandwidth. These two lowpass filters have identical impulse responses, a(t), with transfer functions A(f). The output signals from these two filters are modulated by two carriers, m phase quadrature, but with the same earner frequency fo. The factor ..J2 in Vlcos(2nfor) and Vlsin(2nfor) gives each of these signal a mean-square value of unity [ 10,17 ,35,48]. The output of these two lmear modulators are added to form the QAM signal which is given by;
x(t)
= fi:I.s.,a(t -zT)cos2~J;t + fi:I.s •. ,aCt -zT)sin2~/,t '
.
'
26
The QAM signal is fed to the bandpass filter at the transmitter. This bandpass filter, for practical systems, is very necessary and is used to remove spurious frequency components generated in the modulator [10,17,35,48] It has an impulse response of g(t) with aFounertransfonn ofG(f). The resultmg QAM signal is then transnutted over the transmission path which has been assumed here to be a telephone circmt. The telephone crrcmt mtroduces linear amplitude and phase distortiOn into the transmitted signal, it has been descnbed in Section 2 3 with an impulse response of h(t) and transfer function of H(f) The only additive n01se assumed here is stationary white Gaussian noise, n(t), with zero mean and two Sided power spectral density ~N0 , which added to the QAM signal at the output of the telephone circmt [10-12,17,27,35,48].
18
At the receiving end, the linear demodulator includes at its mput a bandpass filter, the absolute value of whose transfer function matches the amplitude spectrum of the QAM signal at the input to the telephone circmt. The bandpass filter prevents over-Ioadmg of the two following mult1phers by noise in the received signal [48]. Th1s IS performed by removing the noise components outs1de the frequency band of the signal without excessively distortmg the signal Itself. It has an 1mpulse response of c(t) with transfer function of C(f). The output signal from the bandpass f!lteris now coherently demodulated by two reference earners wh1ch have the same frequency but are in phase quadrature. The two lowpass filters after the demodulator suppress the high frequency components so that only the baseband s1gnals are retained. These two Iowpass filters are as those in the transmitter with an impulse response ofb(t) and transferfunctwn ofB(f) [10,17,35,48]. It Will be assumed that the transmitter and rece1ver Iowpass filters are such that;
=
A(f)
=
B(j)
0
lfl > ..t;
2.7
and that ..t; is such that;
1
~
2.8
2T
where 1fT is the Signal-element rate. Defining the complex data symbols, as,
...
where J =
x(r)
H, Eqn =
29
2.6 may be wntten as
"'(se -J2•f,t _1r;;;"--v2 J
+ s , eJ2V,')a (t -11"") I
J
19
...
2.10
-----------------------------------------------------------------------------
where s.' is the complex conJugate of s,. The input signal to the coherent demodulator in Fig 2 3 is now given by
=
m(t)
x(t)
•
g(t)
•
h(t)
•
c(t)
+ n(t) * c(t)
...
211
where *indicates the convolutiOn operation. In the demodulator, the muluplicauon process causes the transfer function of m(t) to be shifted in a negative direction by fc Hz. The lowpass filters b(t) then remove the high frequency components leavmg
only the baseband components. The signal at the output of the two lowpass filters, b(t), are given by;
r 1(t)
=
[Vlm(t)cos(2nfct
+ 6)]
*
b(t)
r 2 (t)
=
[Vlm(t)sin(2nfct
+ 6)]
*
b(t)
...
2.12
213
where 6 is relative phase error between the modulator and demodulator caniers. Combining these two signals in a complex form will give the complex signal r(t)
=
r(t)
r 1(t)
+ j r 2(t)
. .
2.14
consequently, from Eqns. 2.10-2.12
LS1{a(t - iT)*[h(t)*g (t)*c(t)ei2Xt,r}*b (t)
r(t)
I
+ Ls,'{e'••t,ra (t - zT)*[h(t)*g(t)*c(t)ei2XI.r]e'•*b(t) I
+ -V2[(n(t)*c(t))e'""'·'+Ol]*b (I) .. 2.15
LS1y(t- zT) + w(t) I
where
y(t)
{a(t)
*
[(h(t)
* g(t) • c(t))e'~''"]e'' * b(t)
20
2.16
and
w (t)
==
-·~L[_ r;;f(n (t )
*
c (t))e 1 becomes [27]
000
where Er is the transpose of E and u, +•
IS
3.39
a Gaussian random variable with zero
mean and variance
112
= 2criDI 2
000
3.40
Let
r - g----.
n
E.
=
[0
0
0
0
1 0
0
0
0
0]
... 3 41
Now, since the data symbols {s.} are statistically mdependent with zero mean and vmance of 2 c? and since the noise components u, +• and any of the data symbol are statistically independent, it follows that
E[s,s)
=
for
0
i >"j
000
3.42
and
- for
any
j
3.43
Thus, from Eqns. 3.35-3.43 [27]
e
= 2o2IBI 2 + 2criDI 2
where B IS the (n+g+ I)-component row vector given by,
52
...
3.44
-
=
B
E
-
F0
-
E.
...
3 45
It is clear from Eqn. 3.44 that the terms 2821B 1 2 and 2d'l D 1 2 are the mean-square error due to intersymbol mterference and noise, respectively, and the equalizer must be adjusted to minimize both errors. It can be seen from Eqns. 3.44 and 3.45 for any giVen
value
of
J; =e•• , for
D,
2 821B 12
and
e
are
minimized
by
setting
J=I,2, ... g.
Let I be an (n+ I)-square identity matnx and
z. be an (n+ I) upper triangular square
matnx
,, ,,
,,
Yo
0 0
0
'· '·-·
y,
,,
,,_,'·
J,-J
z. ;
0
0 0
0 0 0
'·
345
0 0 0
0
,,
0 0
The (i + I)'• row of
0 0
z.
0
,, ,, ,, y,
0
is derived from the first row by shifting its non-zero
components 1 places to the right, discarding all components after the (n + I)'• and setting the first i components to zero. Fmally, let transpose of
z•.
z; be the complex conjugate
It has been shown [27], that the (n+I)-tap linear feedforward
transversal filter, D, of the deciSlon feedback equalizer of Fig. 3 5 that mimmizes the mean-square error in the equalized signal (Eqn. 3 44), has tap gams given by the (n+ I)-component row vector [27 ,62]
D
=
( E.z:lz.z:
d' + 02 /
)-!
53
...
3.47
~
-------------------------------------------------------------------------------
and the tap gains of the filter F being given by Eqn. 3.34. The matrix
z.z: + ( ~
J
is an (n+ l)x(n+ 1) positve-definite Hermlt!on matrix, bemg posllive-defmite so long as ri' ;e 0 or
z.z; IS non smgular and therefore it has an mverse [63,64]. From Eqns.
3.44 and 3.47, the mean-square error in the equalized signal becomes [27],
...
3.48
3.4.3 Comparison of Equalizers a: In the presence of phase distortion When a channel mtroduces pure phase distortion and IS physically realizable, each pole and zero must be accompanied by a zero or a pole, respectively, located at the complex conjugate of the reciprocals of their values [12]. Therefore, the z-transform of the sampled impulse response of the channel can be represented by
Y(z)
~
Y2(z)y;- 1(z)
...
3.49
where Y2 (z) and Y3 (z) are given by Eqns. 3.27 and 3.31, respectively. Therefore, the z-transform of the filter D of the ZF equalizer is now given by
D (z)
~
z -•y;-'(z )Y3(z)
...
3.50
...
3.51
The z-transform of the channel and filter therefore becomes
Y(z)D(z)
~
z-•
Thus the sampled impulse response of the channel and hnear filter D IS
,--g---.,
n
E
~
[0 0 . . . 0 1 0 . . . 0]
54
3.52
--~
where n and g are ideally infinite but, for practical purposes, can be taken to be appropriately large positive integers. It is clear from Eqn. 3.52 that the tap gains of the filter F are now all zero which means that the decision feedback equalizer has degenerated into a linear equalizerthat achieves the exact equalization of the channel. Furthermore, since the sequence w1th z-transform f 3(z) IS the complex conjugate of the reverse of the sequence wtth z-transform Yiz ), 1t can be shown that the sequence "of the reverse
with z-transform z-"f21(z)Y3(z) is the complex conjugatelof the sequence wtth z-transform Yiz )Y;1(z ), appropriately delayed [27]. Clearly, the filter D is not only the inverse of the channel but IS also matched to the channel, its tap gains being the complex conjugates of the channel sampled impulse response, and m the reverse order. Thus the ZF equalizer is the optimum detector for the given rece1ved signal. When the channel mtroduces pure phase d1stortion, the squared matnx
z. in the
MMSE equahzer, Eqn. 3.46, must ideally have an infmite order [62]. However, a very good approximation to the ideal can ach1eved in practice by a matrix w1th a finite order. It
IS
clear from Eqn. 3.46 that the last column of z.
IS
the sampled
impulse response of the channel, as given by the fust row of z•. but in the reverse order. The square of the unitary length of th1s column is
IYI 2 =
1
3 53
It has been shown that a sequence representmg pure phase distortion IS orthogonal to Itself shifted by any non-zero integral number of places [27], so that the inner product of two sequences is zero. It follows from Eqn. 3.46 that the last column of
z. is for practical purposes orthogonal to each of the other n columns. The factor
E.z; in Eqn. 3.47 is an (n+ 1)-componentrowvector given by the conjugate transpose of the last column of Z., so that E.z;z. IS the (n+ I)-component row vector whose i'h component (fori= 1, 2, .... , n+ 1) is the mner product of the ith and (n + 1)'h columns of z•. This must be approximately zero, except when i=n+ 1, when it is unity (from Eqn 3.53). Thus
E.z:z.
n ~
[0
0
0
0
1] '
= E.
3.54
55
-j
----------------------------------------------------------------------------
and
...
It follows that
E,z;
3.55
is an eigenvector of the matrix z.z;, assoc1ated with an
eigenvalue ofumty [62]. Hence
E.z.·(z.z.• + ci I? I )
~
...
3.56
so thatE.z; is an eigenvector of z.z; + (~J associated with an eigenvalue of [62] 1 +(~).From Eqn. 3 56,
E.z.•
~
(1 + ci1, ·( . ci 82 f.z. z.z. + 82 I
)-I
...
3.57
...
3.58
or
(1
~
ci)-lE.z..
+ 82
which means that E,z; 1s an eigenvectorofthe matrix [z.z; +(~}J associated 1
1
wllh an e1genvalue of[1
+(~]- • From Eqns. 3.47 and 3.58, the tap gains of the
filter D are now given by
56
-
D
(I
r;~)-1
.
+ 02 E,Z,
3.59
As the signal-to-noise rauo becomes extremely htgh, r;l- ---7 0 and D ---7 E,z;, which means that the filter D becomes the same as for the ZF equalizer. When cr'- 1' 0, the tap gains of the MMSE equalizer are [I + ( ~ ]- times those of ZF 1
equalizer, so that the magnitude of the equalized stgnal is reduced by the same factor. This introduces a bias m to the equalized signal such that the real and Imaginary parts of the data component of x, are no longer symmetrically placed with respect to the decision thresholds. Inevitably, this degrades somewhat the performance of the equalizer. b: In the presence of amplitude distortion
In the presence of amplitude distortion it is more difficult to compare theoreucally the performances of the MMSE and ZF equalizers. Thts is essentially because the ZF equalizer is no longer an optimum detector, so that the MMSE equalizer may or may not now have a better tolerance to noise. The fact that the mean-square error in the equalized signal of the MMSE equalizer must be smaller than that m the ZF equalizer, together with the fact that the ZF equalizer has no mtersymbol Interference in the equalized stgnal (with the correct detection of the {s.} ), necessanly implies that the mean-square value of the noise _in the MMSE equalizer is smaller than that in the ZF equalizer. Indeed, since the mean-square error in the MMSE equalizer is caused partly by noise and partly by mtersymbol interference, with a natural tendency for the two be present at rather similar levels, it seems that the equalized Signal of the MMSE equalizer should normally have a sigmficantly lower noise level than that of the ZF equalizer. However, the advantage gained here by the MMSE equalizer is offset by the presence of intersymbol interference m liS equalized signal. The effect of the latter on the tolerance to the nOise is not easy to evaluate theoretically, butts best determined by computer simulation tests. The results presented m chapter 4 and 6 clearly demonstrate this effect.
57
A
(s, l
(x,)
.t.(r,)
Lmear equahzer
Detector
Fig. 3.1 Receiver using linear equalizer
r,
T
T
r, 2
.,.._;:=--··--·······
T
r •-n
.----L------I----I------····--·····--.t._...., L-
x,
'--------------···············----'
Fig. 3.2 Linear feedforward transversal equalizer for the baseband channel
58
A
s,
Detector I
Lmear feedforward transversal f:tlter
Yo
Fig. 3.3 Receiver using nonlinear equalization by decision-directed cancellation of intersymbol interference
r, Yo
A
s,
x,
Detector
A
A
T
T
T
Yo
Fig. 3.4 Detector and pure nonlinear equalizer
59
A
r1
T
r1.1
T
r1-2
T
r 1-n
X
St-n
1
Detector
q1
dn
X
A
A
A
S1-n-g
St-n-2
St-n-1
T
T
"'
T
0
Linear filter D
Lmrear filter F
----------------
Fig. 3.5 Decision feedback equalizer
-----------------
Channel! Real 10000 0 5031 -0 1447 00300 00094 -0 0136 0 0102 -0.0108 00083 -00044 00001 00014 -0.0005 -0 0007 00000 00004 -00002 -0 0002 -00002 00000 00000 00000
Channel2
Imagmary 00000 02008 -0.0083 -0 0097 00077 -0 0039
1.0000 0.5006 -0.1678 0.0176 -0 0062 -0 0146
00006 -0 0013 00024
00080 -0 0049 00063 -0 0031 00013 00012 -0 0003 -00011 0.0004 00008 -00002 00004 -0 0002 00004 00001 00000
-0 0020 00024 -0 0021 00019 -0 0008 -0.0002 00006 -0 0004 -0.0003 00004 0.0003 0.0000 00000
10000 04608 -0.5824
00007 -0 0011 -00002 00003 00000 00003 00000
00013 -0 0016 -0.0003 -0 0001
----
----
-------
----
----
----
----
----
-------
----
----
----
Real
00000 0.3397 00282 -0.0391 00229 -0 0132 00012 -00004 -0.0001 00027 -00020 -00003 00001 00002 -00008
----
----
Table 3.1
Imagmary
----
----
-------
Real
Channel3
0.1573 -0 0175 -0 0021 -0 0021 -0 0051 00080 -0 0039 -0 0001 00038 -0 0009 00023 -0.0004
00003 -0 0007 -0.0008 00001 -0 0001 -0 0001 00000
----
Channel4
Imagmary 00000 1 1004 00436 -0 1729 00872 -0 0194 00083 -0 0075 00054 -0 0035 00014 -0.0056 00026 -0 0027 -0 0009 00018 -0 0013 00000 00001 00002 00011 -00003 00002 00007 00004 0.0001
----
Real
Imaginary
10000 02458 -1.7189 06743 00356 -0.1155 00296 -0 0168 00160 -0 0146 -0.0002 00020 -0.0003 -0 0025 00066 -0 0035 -0 0061
00000 1.9797 -0 2028 -0 7923 05062 -0 1449
00059 -0 0008 -0 0006 -0.0013 00010 00015 -0 0003 -0 0001
-0 0009 0 0038 -0 0002 -0 0003 -0 0024 0 0009 0 0005 -0 0001
00000 00000
00000 00000
00385 -0 0370 00185 -0 0011 -0 0029 -0 0040 0.0002 -0 0074 0 0044 00050 -0 0028
Sampled Impulse responses of the mimmum phase versions of channels
1-4.
61
Channel6
ChannelS Real
Imagmary
1.000000 1.346393 -0 148268 -0222296 0.200053 -0.139489 -0.029114 0 040268 -0 006336 -0 005842 0022487 -0 020116 0010094 -0 000049 -0 014825 0022883 -0 024204 0 017227 -0 005489 -0 007162 0.015331 -0 013887 0 007161 0 001364 -0 004231 0 005047 -0 003534 -0.004201 0.000481 0000392 0000055
0000000 0 359591 0 330302 -0 087503 0 002161 0036076 -0 019378 -0 011521 0 019953 -0 010060 -0005228 0018909 -0 021531 0022504 -0 019873 0.008106 0003678 -0 014030 0.018100 -0 016173 0 008199 0 000381 -0 008364 0 009576 -0.005552 -0 001576 0004075 -0002520 -0 001848 -0.000019 0000103
----
-------------
----------
----
-------------
-------
Table 3.2
----------------------
Real 1000000 1.173137 -0.169015 -0.138032 0 199112 -0 117248 -0004179 0 043908 -0 021240 0 012810 -0006241 0002090 -0.000932 -0002066 0001822 -0 001417 0.001212 -0000042 -0000445 -0 001481 0 001702 0003671 0.004560 -0011131 -0 008303 -0 003463 -0 000987 -0 000218 -0000024
-------------
-------
----------------------
Channel?
Imagmary 0000000 0202423 0.108631 -0 049378 0 027682 0.007974 -0 009518 0006603 0001454 0002679 -0 004201 0.002607 0 003301 -0 005158 0 002641 0.001506 -0 003261 0002869 0 000283 -0.004083 0 002728 0002814 -0 002912 -0 000303 -0.000210 0 000532 0000288 0 000112 0000016
----
----------------
----------------
-------
Real 1000000 1.220912 -0 849086 -0452288 0482880 -0 256621 0 000383 0 096231 -0.070512 0024782 -0 002350 -0 006643 0005802 -0 004082 0003905 0 002613 -0 001821 0 007187 -0 003219 0.003162 -0 000893 -0 001047 0000104 -0 002343 0002986 -0 002837 0001167 0000222 -0003119 0 002629 -0.001428 -0 002190 0002643 -0 000528 -0 002267 0 002710 0 001723 -0 001346 -0 002851 -0 001450 -0 000571 -0000072
Imagmary 0000000 1339438 I 089636 -0 648090 0 037754 0212632 -0 201586 0 091226 0.008677 -0 021441 0 016130 -0 003623 -0 001010 -0 000398 -0 005003 0002401 -0 003081 -0 002388 0 002566 -0 003157 0002685 -0 001343 0 000071 -0000349 0 000001 0 001728 -0 000987 0 001269 -0 000076 -0 000807 0 001386 0000206 -0 000385 0000282 -0 000265 -0 000468 0 000499 -0 000249 -0 001539 -0 001316 -0 000521 -0 000038
ChannelS Real 1000000 1416469 -3 199131 -2 316267 2442584 -0.246997 -0 681720 0621926 -0252233 -0 054037 0.091835 -0.040897 -0.003828 0 016616 -0 008264 -0.001113 -0 004185 -0 000335 0 016538 -0 000591 -0 007670 -0 001936 -0 000837 -0 000885 0.000102 -0.000102 0000088
·---
----------------
----------------------
-------
Imagmary 0 000000 2 774325 2.788691 -2 920049 -0 938336 I 629715 -0 732925 -0005074 0293966 -0 203288 0 044110 0032256 -0 027159 -0 003513 0 000749 -0 013205 0 001288 -0 006626 0 005191 0 012238 0 002827 0002042 -0 000467 -0.000958 -0 000414 0 000190 0 000130
----------
-------------------
----------
-------
----
Sampled impulse responses of the minimum phase versions of channels
5-8.
62
CHAPTER4
ADJUSTMENT OF THE PRE-DETECTION FILTER
4.1 INTRODUCTION This chapter is concerned with the adaptive adjustment of an advanced data receiver, when operating in the presence of signal distortiOn and additive noise. For channels introducing severe degree of bandlimiting, coupled with the requirement of a high transrmssion rate, a s1gmficantly better performance can be ach1eved by employing a linear f1lter, called pre-detection filter, ahead of the detector [27,29,32,35,61] as shown m Flg. 4.1. The function of the filter is to concentrate the energy of the sampled impulse response of the channel and filter such that it appears towards the earlier samples [56]. Furthermore, it rmmm1zes (according to some cnterion) the components (pre-cursors), proceedmg the largest component m the sampled impulse response of the channel and filter which cause mtersymbol interference to future data symbols [56,61]. Thepre-detectionfiltercan also be combined with anon-linear equalizer (Flg. 4.1), wh1ch reqmres the knowledge of an estimate of the sampled 1mpulse response of the channel and pre-detection filter, to remove the mtersymbol interference caused by previously detected data symbols This chapter investigates several methods to actually ach1eve the adaptive adjustment of the pre-detection filter by studying two adjustment cntena, that of max1mizing the s1gnal-to-noise ratio at the detector input when the tap-coefficients are g1ven by the last (n+ !)-coefficients of the infmite set of values used m the 1deal filter (where n is an appropriate integer) and secondly, the mmimum mean-square error criterion. Forthe adJustment of the f1lter according to the first cnterion, three different schemes have been presented. The linear feedforward transversal pre-detection filter is, here, an all-pass network with ideally an infinitely long sampled impulse response, that adjusts the combined sampled impulse response of the channel and f1lter to be minimum phase w!lhout, however, changing any amplitude distortion introduced
63
by the channel and without changing the Signal to noise ratio at the output of the filter, compared to that at its input [27]. The pre-detection filter can, forconvemence and as an aid to understandmg, be considered to operate in two stages. Firstly, It equalizes all phase distortiOn introduced by the channel to give a resultant sampled impulse response that IS lmear phase m character. Secondly, the filter converts the linear phase response into a minimum phase response which results m a sampled impulse response that has a rapid nse to its peak value, followed by a rapid decay, with relatively few post-cursor lobes; in essence, this ensures that the energy of the sampled impulse response of the system is concentrated towards the earher samples with especial emphasis on the first sample, without, however, changmg the signal to noise ratio at the output of the filter [35]. All three schemes require a knowledge of the roots of the z-transform of the sampled impulse response of the channel that he outside the umt Circle in the z-plane. These roots are then used to determine the tap-coefficients of the pre-detection filter such that the cascade of the sampled impulse response of the channel and filter has a z-transform whose roots lie ms1de the umt crrcle. The schemes differ by the mechanism used to determine the coefficients of the pre-detection filter, wh1ch for perfect adjustment, will be infimte in number Clearly, in practice, the number of taps must be restricted to the smallest number consistent with adequate equalizatiOn. The fourth and the final scheme presented differs from the first three schemes in that the adjustment cnterion attempts to minimize the mean-square error m the equalised signal. Computer simulation studies when operating over models of eight different telephone channels are presented for various schemes. The results are arranged to show the comparative accuracies and complexities (m terms of the number of arithmetic operations required to Implement each scheme). Furthermore, the results have been arranged to Illustrate the behaviour of the adJUStment schemes m the presence of addmve white Gaussmn nmse and different amount of distortion introduced by the channels, as well as the system dependence upon the length of the linear pre-detection filter.
64
4.2 ADJUSTMENT SCHEME 1 Let Y(z) be the z-transform of the sampled Impulse response of the lmear baseband channel, given by
Y(z)
...
where {u,} are the negative of the g roots (zeros) of Y(z), such that Y(z
4.1
=-u,) =0.
Y(z) can also be written as ...
4.2
...
4.3
...
4.4
where
with all zeros lying within the umt circle in the z-plane, and
with all Its k zeros lymg outside the unit crrcle. Thepre-detection filter, D, in Fig. 4.1, is a Iinearfeedforward transversal filter whose tap-coefficients are given by the (n+ 1)-component vector D
=
[d0
d1
•
•
•
dJ
...
4.5
...
4.6
with z-transform
65
The resultant sampled impulse response of the channel and filter is therefore the (n+g+ I)-component vector,
...
4.7
...
48
whose z-transform is, E(z)
=
Y(z)D(z)
+ . . . + en+g z-59, revealed no significant improvement in the values of 'Jf1 and 'Jf2 but mvolved a large increase in the number of arithmel!c operations mvolved. Finally, the values of the mean-square error in the equalized signal due to the presence of additive white Gaussian noise, together with the overall mean-square error introduced by the combined effect of addl!ive nmse and mtersymbol interference, have been computed. The mean-square error due to the addltlon of noise only is taken to be given by,
4.20
where D is as defined in Eqn 4.5 but where D is smtably scaled such that e. is equal to unity, and where 2r:f is the variance of the (complex) noise process, w,. Tables
69
4.7 and 4 8 show the results of these tests for various values of signal-to-noise ratio; the results were found to be independent of (n+ 1), the number of taps in the hnear filter D. The overall mean-square error present in the signal at the detector input, x, due to both additive noise and intersymbol interference is given by,
e =
IO!og10 (
1 20000 .~ lx, 20000 1
- s,_,i
2)
4.21
In tests fore, it is assumed that the signal passmg through the filter F (Fig. 4.1) is s, mstead of s,. The value of e for various conditions was obtained from computer Simulauon tests involving the traimng sequence of20,000 symbols; these values for different signal-to-noise ratios are tabulated in Tables 4.9-4.16, for values of n set to 29, 39, 49 and 59.
4.3 ADJUSTMENT SCHEME 2 As with scheme 1, this scheme attempts to determme the pre-detection filter tap-coefficients such that the sampled Impulse response of the channel and filter is minimum phase. Furthermore, the tap-coefficients, {d.}, of the filter depend only on the kroots ofY(z) thatlie outside the unitctrcle in the z-plane. Having determmed these roots (by using some suitable root-findmg algorithm), the adjustment scheme forms, as with scheme 1, the polynomials Y2(z) and Y3 (z), where Y2(z)
= Y2.o +
Y2.1z
Y3(z)
=
YJ,Iz
-1
+
+
Y2.kz
+
YJ,kz
-k
...
4.22
...
4.23
and
YJ,o
+
-I
+
Also it forms the polynomta!
70
. .
-k
==
.
Y. o + Y..,z
+ • · · + Y. _.
30
-18 54
-18 80
-15 47
-10 27
40
-28 54
-28 80
-25 47
-2027
50
-3854
-38 80
-35 47
-3027
60
-48 54
-48 80
-45 47
-4027
70
-58 54
-58 80
-5547
-5027
80
-68 54
-68 80
-65 47
-6{)27
Table4 7
Mean-square error m the equahzed s1gnal due to add1Uve wh1te Gausswn noise only usmg scheme I.
sr-..'R
ChannelS
Channel6
O!annel7
Cllannel8
20
254
070
8 22
18 41
30
-746
-9 30
-I 78
8 41
40
-17 46
-19 30
-11 78
-I 59 -1159
50
-27 46
-2930
-21 78
60
-37 46
-3930
-31 78
-2159
70
-4746
-49 30
-41 78
-31 59
80
-57 46
-5930
-51 78
-41 59
Table4 8
Mean-square error m the equahzcd s1gnal due to add1Uve whlte Gauss1an nmse only usmg scheme I.
85
----------------------------------------------------------------------------------------------
sl\ra
Channel!
Channel2
Channe13
Channel4
20
-8 58
-8 83
-5 51
-028
30
-18 52
-18 78
-15 45
-9 86
40
-28 52
-28 78
-25 45
-17 35
50
-38 55
-38 80
-35 44
-2007
60
-48 53
-4875
-4513
-2056
70
-58 47
-58 31
-5297
-2057
80
-'R
Channel!
Channel2
Channcl3
Channel4
20
-8 58
-8 83
-5 51
-() 30
30
-18 53
-18 78
-15 45
-10 21
40
-2853
-28 78
-25 46
-19 92
50
-38 55
-38 81
-35 48
-2771
60
-48 54
-4880
-45 47
-30 81
70
-58 53
-58 79
-55 46
-31 22
80
-68 55
-68 80
-65 48
-31 25
Table 4 13
Mean-square error m the equaliZed s1gnal w1th 50-taps m the hncar filter D, usmg scheme L
SI\'R
ChannelS
20
258
Channel6 072
Channel?
ChannelS
8 73
18 44
30
-7 29
-929
167
854
40
-15 91
-19 12
-() 61
-() 36
50
-2039
-2825
-() 88
-544
60
-2129
-33 66
-() 91
-6 51
70
-2140
-34 83
-() 95
-667
80
-2132
-34 95
-() 80
-652
Table 4 14
Mean-square error m the equal !Zed signal With 50-taps m the hnear filter D, usmg scheme I
87
SNR
Channel I
0Jannel2
Channel3
Channel4
20
-865
-8 89
-605
-3 03
30
-18 57
-18 83
-IS 57
-11 15
40
-28 54
-28 82
-25 49
-2037
50
-38 55
-38 80
-35 50
-30 19
60
-48 63
-48 88
-45 56
-39 89
70
-58 56
-58 84
-5549
-49
80
-68 58
-68 85
-65 51
-58 59
Table 4_15
os
Mean-square error m the equaliZed s1gnal with 60-taps m the lmear filter D, usmg scheme L
Sl\'R
Channel 5
Channel6
Channel7
ChannelS
20
2 57
072
8 39
18 44
30
-7 38
-930
-t 26
8 43
40
-16 77
-19 23
-8 62
-I 44
50
-23 26
-2925
-11 21
-1079
60
-2512
-3925
-11 58
-1711
70
-25 31
-4860
-1165
-18 79
80
-25 39
-54 81
-!I 52
-18 85
Table 4 16
Mean-square error m the equaliZed s1gnal w1th 60-taps m the lmear filter D, usmg scheme L
Cbarme12
Channel I
Channel3
Channel4
n+l
Add.& Sub
MulL
20
1077
708
1106
805
1106
805
1292
913
30
2342
1786
2371
1785
2371
1785
2557
1893
40
4107
3148
4136
3156
4136
3156
4322
3273
50
6372
4928
6401
4945
6401
4945
6587
5053
60
9137
7108
9166
7125
9166
7125
9352
7233
Table 4.17
MUiL
Add. & Sub
Add.& Sub
MulL
Number of anthmet1c operat1ons mvolved m scheme 2
88
Add& Sub
Mult
Channel6
ChannelS
ChannelS
Channel?
n+1
Add. &Sub
Mult
Add. & Sub
Mull
Add.& Sub
Mull
Add. & Sub
Mull
20
1292 2557
913 1893
1356 2621
950
1590 2855
1085
1682 2947
1138
4322
3273
4386
50
6587
60
9352
5053 7233
6651 9416
30 40
Table 4 18
1930 3310 5091
6885
7270
9650
4620
2065 3445
4712
2118 3498
5225
6977
5278
7405
9742
7458
Number of an!hmetic operatiOns mvolved m scheme 2.
Channell
Channel2
Channe13
Channel4
n+1
Add.& Sub
Mull
Add.& Sub
Mull
Add & Sub
MulL
Add.& Sub
Mull
20
735
492
980 1340 1780
656
980 1340
656 976
1960
1312
976
1780 2220
1296 1616
2680 3560 4440
1952 2592 3232
2900
1936
5800
3872
30
so
1005 1335 1665
732 972 1212
2220
1296 1616
60
2175
1452
2900
1936
40
Table4.19
Number of an!hmeuc operatiOns mvo!ved m scheme 3.
ChannelS
Channe16
Channel?
ChannelS
n+1
Add. &Sub
Mull
Add.& Sub
Mull
Add. & Sub
Mull
Add.& Sub
Mull
20 30 40
1808 2680
1312 1952
1804 2684
5335
3564
2925 4355 5785
2132
2592
1476 2196 2916
2695 4015
3560
2025 3015 4005
3172 4212
4440 5328
3232 3872
5095 6984
3636
6655 7975
4444 5324
7215 8645
5252 6292
50
60
Table4 20
4356
Number of anthmeuc operatiOns mvolved m scheme 3
89
Channel}
Channel2
Channel4
Charmel3
S~'R
w,
w,
w,
w,
w,
w,
w,
w, -{) 18
20
-3908
-3379
-39 95
-34 59
-27 48
-17 18
-1972
30
-58 56
-53 53
-59 41
-54 36
-43 89
-33 76
-3053
-9 31
40
-78 14
-73 52
-7678
-1514
-6278
-53 95
-43 56
-28 36
50
-88 57
-85 93
-8019
-65 28
-7248
-53 56
-50 14
-1255
60
-8903
-6387
-8030
-4515
-73 02
-33 20
-6069
-3 61
70
-8911
-43 92
-8084
-25 69
-74 52
-1470
-7079
-{) 91
80
-8976
-2457
-8093
-978
-8275
-2 94
-7736
252
Table4 21
Discrepancy between the mimmum mean-square error sequence and the mmimum phase sequence with 30-taps m the filter D.
ChannelS
SNR
Channel7
Channel6
w.
w,
w.
w,
Charmel8
w,
w.
w,
w,
20
-2005
-2 84
-2054
-5 98
-13 93
3 94
-11 02
1613
30
-29 05
-7 83
·29 4t
·15 22
-2136
-t 36
-t5 15
1449
40
-38 60
-15 08
-37 02
-14 67
-3090
-12 57
-2t 98
tl54
50
-4862
-2t 04
-47 16
-523
-4099
-8 83
-29 59
576
60
-5975
-t217
-57 32
-{) 87
-4995
-t 41
-3873
-tl 59
70
-6892
-736
-67 t7
I 19
-59 13
3 69
-4516
7 48
80
-7797
-4 93
-7766
309
-68 24
6 64
-54 81
14 99
Table4.22
Discrepancy between the mimmum mean-square error sequence and the mmimum phase sequence With 30-taps m the filter D. Olannell
Channel2
Channe13
Channel4
S/'."R
w,
w,
w,
w,
w,
w,
w,
w,
20
-39 07
-3379
-3995
-3459
-2749
-17 17
-19 72
-{) 18
30
-58 56
-53 53
-59 44
-54 34
-43 85
-33 58
-3055
-9 29
40
-78 51
-73 50
-79 38
-74 32
-63 10
-52 89
-44 89
-25 29
50
-9847
-93 51
-99 05
-94 47
-8292
-7286
-52 91
-19 59
60
-t16 07
-liS t5
-109 88
-t07 08
-9810
-9056
-61 57
-8 17
70
-11968
-105 50
-tlO 39
-85 42
-9978
-6702
-7229
-3 66
80
-tl9 74
-8525
-tlO 40
-65 41
-99 84
-47 98
-8665
-2 tl
Table4 23
Discrepancy between the mimmum mean-square error sequence and the mimmum phase sequence with 40-taps m the filter D.
90
Channel6
ChannelS
SNR
'V,
20
-20 06
-2 33
30
-29 36
.673
40
-39 61
50
ChannelS
Channel?
'V,
'1/,
'V,
-2070
-595
-13 93
3 94
-11 02
16 13
-3067
-13 82
-2136
-136
-15 19
14 49
-1241
-43 94
-26 85
-3090
-12.57
-22 21
11 68
-50 81
-2145
-52.85
-29 12
-4099
-8 83
-3073
6 84
'V,
'1/,
"'·
60
-5771
-18 91
-58 90
-13 40
-49 95
-I 41
-4020
-439
70
-69 rn
-965
.6515
-4 93
-59 13
3 69
-4825
0 85
80
-8159
.653
-77 80
.039
.68 24
664
-56 82
10 10
D1screpancy between !he mmimum mean-square error sequence and !he m1mmum
Table4.24
pbase sequence w11h 40-taps m !he filter D.
Channel2
Channell
S!\'R
'1/,
Channel4
Channel3
'V,
'V,
'V,
'1/,
'V,
'1/,
20
-39 rn
-33 79
-39 95
-3459
-27 50
-17 17
-1972
.0 18
30
-5856
-53 53
-59 44
-54 34
-43 84
-33 55
-3055
-929
40
-78 51
-73 50
-79 39
-74 31
.6306
-52 83
-45 76
-2419
50
-98 50
-93 50
-99 38
-94 31
-8217
-72 75
-59 81
-44 21
60
-118 5 0
-113 50
-119 35
-114 32
-102 94
-92 80
-64 53
-19 85
70
-138 2 t
-13373
-13691
-135 67
-12144
-115 60
-71 fl7
-1023
80
-1497 5
-145 29
-140 50
-125 65
-12665
-100 88
-81 07
-4 01
Table4 25
Dtscrepancy between !he mmtmum mean-square error sequence and !he mmimum pbase sequence WIIh SO-taps m !he filter D.
ChannelS
S!\'R
'V,
20
-20 06
Channel6
Channel?
ChannelS
'V,
'1/,
'V,
'1/,
'V,
'V,
-233
-2070
-5 95
-13 98
392
-1102
1613
30
-30 00
.672
-3067
-13 82
-21 71
-I 11
-1519
14 49
40
-4008
-12.14
-4425
-2649
-3171
-9 31
-22.22
1168
50
-5126
-20 65
.6027
-48 90
-4154
-17 42
-3097
696
60
-641 4
-4005
.64 83
-3167
-51 28
-4 73
-41 40
-150
70
-70 22
-21 38
.6715
-13 69
.62.39
.052
-5164
-15 99
80
-77 83
-1141
-76 31
-3 74
.6844
2 00
-59 24
3 12
Table4 26
Discrepancy between !he mm1mum mean-square error sequence and !he mmimum pbase sequence With SO-taps m !he filter D.
91
Channel2
ChaMell
Channel3
SNR
'If,
IV,
IV,
IV,
20
-39rrl
-33 79
-3995
-3459
30
-5856
-5353
-59 44
40
-78 51
-7350
-7939
50
-9850
-93 50
-99 39
60
-118 50
-113 50
70
-138 48
-133 58
80
-159 06
-153 65
Channe14
'If.
IV,
IV,
'If,
-27 50
-1717
-19 72
-0 18
-5434
-43 84
-33 54
-3055
-9 29
-7432
-6305
-52_81
-45 81
-2414
-94 31
-8296
-7273
-62 34
-5074
-11938
-11431
-10295
-9272
-66 88
-2727
-139 37
-134 34
-122 94
-112 72
-7286
-13 06
-159 67
-15147
-142 67
-133 96
-83 21
-7.58
Discrepancy between !he mmimum mean-square error sequence and !he mimmum phase sequence with 60-taps m !he filter D
Table4 27
ChaMelS
Channe16
Channel?
Cha!Ulel8
SNR
'If,
IV,
'If,
IV,
IV,
IV,
IV,
20
-2006
-233
-2070
-595
-13 98
392
-11 02
16.13
30
-29 37
-672
-3067
-13 82
-21 75
-I 10
-15 19
1449
40
-4009
-1211
-4429
-26.46
-3275
-8 53
-2223
1168
50
-5158
-19 78
-6219
-4438
-45 05
-2055
-3098
697
60
-6447
-3708
-80 68
-6697
-5514
-2074
-4177
-I 21
70
-73 05
-2733
-86 63
-54 91
-62 44
-912
-54 89
-16 92
80
-8214
-17 11
-87 04
-34 87
-7199
-265
-64 84
-!I 73
Table4 28
SNR
IV,
Discrepancy between !he mmimum mean-square error sequence and !he mimmum phase sequence wah 60-taps m !he filter D_
Channell
Channel2 -
Channe13
Channel4
20
-8 69
-8 93
-610
-3 08
30
-18 55
-18 81
-15 55
-1111
40
-2853
-2879
-25 46
-19 87
50
-3857
-38 82
-35 47
-2745
60
-4854
-4874
-45 08
-3799
70
-5842
-58 11
-53 19
-4727
80
-67 63
-66.15
-61 61
-5430
Table4.29
Mean-square error m the equalised Signal usmg scheme 4 with 30-taps in !he linear filter D
92
SNR
ChannelS
Channel6
Channel7
Channel 8
20
1.51
064
673
9 38
30
-7 82
-823
-078
3 61
40
-16 55
-15 86
-7 47
-3 23
50
-25 81
-2527
-16 84
-4 12
60
-3575
-34 01
-17 73
-8 63
70
-4409
-4341
-2714
-8 00
80
-5144
-53 90
-3690
-19 41
Table4.30
SNR
Mean-square error m the equahsed Signal usmg scheme 4 With 30-taps in the lmear fllterD
ChaMell
Channe12
Channe13
Channel4
20
-8 69
-8 93
-610
-3 08
30
-18 56
-18 81
-15 56
-1113
40
-2853
-2879
-25 47
-2029
50
-38 58
-38 83
-35 51
-29 43
60
-48 58
-48 81
-45 49
-38 85
70
-58 55
-58 80
-55 47
-4798
80
-6854
-6879
-6539
-57 58
Table 4.31
Mean-square error in the equaliSed Signal usmg scheme 4 with 40-taps in the hnear fllterD
SNR
ChannelS
Channel6
20
I 46
054
30
-797
-899
-I 51
3 41
40
-1724
-18 89
-977
-3 66
50
-2693
-2815
-19 00
-9 81
-
Charmcl7
CbannelS
6 54
9 38
60
-3515
-36 01
-2574
-15 17
70
-45 32
-4314
-34 53
-14 68
80
-55 40
-5444
-41 28
-2177
Table4 32
Mean-square error m the equalised Signal usmg scheme 4 With 40-taps m the lmear filter D_
93
SNR
Channel!
Channel2
Channe13
20
-8 69
-8 94
-610
-3 I
30
-18 56
-18 82
-15 56
-1113
40
-28 53
-28 79
-25 47
-2040
50
-38 58
-38 83
-35 51
-3016 -39 30
Channel4
60
-48 56
-48 81
-45 49
70
-58 55
-58 80
-55 48
-47 87
80
-68 54
-68 80
-65 54
-5710
Table4 33
Mean-square error m !he equalised Signal using scheme 4 wi!h 50-taps m !he lmear filler D.
S~'R
OtannelS
Cbannel6
Channel?
ChannelS
20
I 46
054
6 50
9 37
30
-798
-8 99
-I 83
3 41
40
-17 43
-18 97
-10 17
-3 75
50
-2713
-2909
-18 57
-11 81
60
3719
-3806
-27 49
-18 94
70
-45 53
-4445
-3644
-3002
80
-53 98
-54 21
-4123
-3145
Table4 34
S~'R
Mean-square error m !he equalised Signalusmg scheme 4 wilh 50-taps m !he lmear filter D.
Channel I
Channel2
Channcl3
-
Channel4
20
-8 69
-8 94
-611
-3 09
30
-18 56
-18 82
-15 56
-1113
40
-2854
-2879
-25 47
-20 41
50
-38 58
-38 84
-35 51
-3027
60
-48 56
-48 82
-45 49
-39 83
70
-58 55
-58 81
-55 48
-4907
80
-68 54
-68 80
-65 47
-58 65
Table4.35
Mean-square error m !he equalised Signal usmg scheme 4 with 60-taps m !he hnear filterD
94
S~"R
ChannelS
Charmel6
Ch:umcl7
ChannelS
20
I 46
054
650
9 37
30
-7 98
-8 99
-I 88
3 41
40
-17 44
-18 98
-11 25
-375
50
-2723
-2923
-2065
-11 89
60
-3728
-39 38
-28 50
-21
70
-46 89
-4924
-35 56
-28 79
80
-56 61
-5874
-43 90
-33 26
Table4 36
r:n
Mean-square error m !he equal1sed signal usmg scheme 4 wllh 60-taps m !he lmear filter D.
95
---------------------------------------------------------------------------------
CHAPTERS
ALGORITHMS FOR THE ADJUSTMENT OF THE PRE-DETECTION FILTER
5.1 INTRODUCTION Polynomials are of fundamental importance in numencal methods because many functions or systems niay be approximated by them [65-66]. The z-transform of the samples y0, Y( z )
y 1, =
•
•
•
,
y, IS the polynomml
-1
Yo + y 1z
+ . . . + y8 z-g
...
51
This polynomial is useful and it has several properties [65-71] given below 1-
A g-degree polynomial (Eqn. 5 1) will have g roots (zeros). These roots are the solution of the polynomial when equated to zero and they may be real or complex.
n-
If all the {y.} coefficients are real, then all complex roots will appears m complex conjugate pairs.
1ii-
When a zero is found, it can be used to reduce the degree of the polynomial by removmg It from the polynomial.
iv-
Polynomials provide a good example of the use of the mvanance principle because three different transformations can be carried out on polynomials leaving them essentially unchanged as given below [66] (1)- the transformation ofY(z) into cY(z), where c IS an appropriate constant,
(2)- the transformation of Y(z) into Y(cz),
96
-----------------------------------------------------------------
(3)- the transformation of Y(z) into z"Y(z ). Numerical methods for finding and identification of the roots (zeros) of a polynomial are often required in analysis , design or implementation of various branches of applied mathematics. It is relatively easy to find algonthms for computing the roots of polynomials With real-valued coefficients , but It is much more difficult to find algorithms which solve the more complicated problem associated with complex coefficients of high degree (greater than 20) polynomials. The problem changes from one searching for roots along the real axis or complex conjugate root pairs to one searching the complex plane for the desired roots [66]. One method to obtain an approximate value of the root is to plot the function (impulse response) and determine where it crosses the z axis. The value of z for which f(z)=O represents the root. Although graphical methods are useful for obtaining rough estimate of roots, they are limited because of their lack of prec!Slon and application. Alternative methods using numerical approach rather than graphical, have been considered m th1s chapter. The roots required for the adjustment of the pre-detection filter used m this work and descnbed m Sections 3.4.1 and 4.2-4.4 are those with absolute values greater than unity. However, most of the algonthms m the literature have been found dealmg with all the roots (inside and outside the unit circle m the z-plane) except a few algonthms which have been mentioned m references 39 and 72 which deal with the reqmred roots. This chapter presents algorithms for locating or identifymg the required roots. It also compares them. from the point of view of accuracy and complexity. The first group of algorithms identify the roots with absolute values greater than unity, whereas the second group of algorithms attempt to locate those roots.
97
5.2 ROOT-IDENTIFICATION ALGORITHMS 5.2.1 Schur Algorithm This algorithm is based on a criterion suggested by Schur [72-74] which may be used to determine whether or not a given polynormal has a root lying outside or inside a given circle in the complex z-plane. This can be achieved by a particular lmear transformation. Let
f(z)
...
=
where {a.} are the complex valued sample values ofthepolynomtal andf(O)
52
;t
0.
Also let/(z) be a polynomial whose coefficients are the complex conjugate of the reverse of those of f(z), such that
/(z)
=
a:
+ ... +
+
...
5.3
...
5.4
The particular lmear transformation that has been suggested IS
T'(f(z)]
=
a;f(z)
-
a./(z)
Th1s combination results in a sequence of polynormals m an order of decreasmg degree. The constant term ofT'(f(z)] is T'(f(O)] where
T'[f(O)]
...
5.5
This constant term is of a particular mterest because It is real-valued. Furthermore, this constant decides whether f(z) is a fit polynomial to apply the transformation g1ven by Eqn. 5.4. It also decides whether or not f(z) has a root inside the unit circle in the z-plane according to the basic theorem [71-73];
98
If for some h>O, r•[f(O)] < 0, then f(z) has at least one root mside the unit circle. If
instead, T'[f(O)] >0 for integer for which r•(f(O)]
1 $i O), r•[f(O)] < 0 or the degree of th~ reduced polynomial reaches 1, with the remaimng constant greater than zero.
This procedure IS easily understood using the flow diagram shown in Fig 5 1. Cons1der now the z-transform Y(z) of the sampled impulse response of a linear baseband channel given in Eqn. 5.1. Y(z) IS a polynomial in z-1 of g degree, and can be expressed as a function of z- 1, f(z- 1). It has g roots, k of which have a absolute values greater than unity. These k roots are those roots of f(z- 1) whose magnitudes are less than umty. The algorithm given in Eqn. 5.4 will now be applied directly on
99
----------------------------------------------------------------------
f(z- 1)> to g1ve a decision whether f(z- 1) has a root lying ms1de the unit crrcle in the
z-plane. This eventually suggests that whether Y(z) has a root lying outside the unit circle in the z-plane or not. The algorithm has been tested by computer simulation over eight different models of telephone channels referred as channels 1-8. The sampled Impulse responses of these channels are as given in Tables 2.1 and 2 2. The algorithm has also been tested over the minimum phase version of channels 1-8, pven m Tables 3.1 and 3.2, where these channels have no roots lymg outside the unit circle in the z-plane. The most Important requirement m this study IS the accurate knowledge of the roots of the z-transform of the sampled impulse response of channels 1-8 that lie outside the unit circle in the z-plane. These roots were located usmg appropriate NAG (Numerical Algorithm Group) routme algorithm [60] runmngon a Honeywell DPS-8 computer; these roots along with their absolute values, are g1ven in Tables 4.1 and 4.2, respectively for channels 1-4 and 5-8. Throughout the tests the number of real arithmetic operations involved in the algorithm to give the required decision were calculated and recorded as shown in Tables 5.1-5.8. Furthermore, from the tests, the following interesting points have been found. 1-
The algonthm did not failed with any of the channels tested. The number of arithmetic operations required for each channel to give the right decision are as shown in Table 5.1 and 5.2, respectively, for channels 1-4 and 5-8.
u-
The value of T"[Y(O)] decreases ash increases, and after a certain value of h ,11 becomes very small (approx1mately zero). This major problem appeared because of the transformation given by Eqn 5.4, where at each step Y(z) or its successive modified version, has to be multiplied by y0 and y8 , bearing in mmd that these two components are less than unity This problem has been solved by checking at each step of transformation T"[Y(O)]. Whenever this component becomes less than 10-5 , all the coefficients ofT"[Y(z)] should be multiplied by 105• Th1s of course has no effect on the decision considered by the algorithm.
100
ii1-
Dropping the lower-degree components ofY(z) has no effect on the decision considered by the algonthm. Furthermore, it reduces the number of arithmetic operations involved to give the decision especially 1f the channel is minimum phase as shown m Tables 5.3-5 8.
1v-
All the minimum phase channels tested produce the same results in terms of the number of arithmetic operations involved when the number of the components in the sampled impulse response of the channel are the same. Therefore, the results of one minimum phase channel are considered here to represent all the minimum phase channels as shown in Table 5.8, where the number of components in the sampled impulse response of the channel1s the only factor which affects the number of mthmetic operations involved.
5.2.2 Nyquist Criterion The Nyqmst criterion is an analysis tool for determmmg whether or not an impulse response is mimmum phase As mentioned earlier, the impulse response of the channel is minimum phase only 1f all the roots (zeros) of the z-transform of the impulse response he ms1de the umt crrcle m the z-plane Therefore, a contour which encloses the entrre roots that lie outside the circle in the z-plane must be chosen, determimng whether any of those roots lie withm that contour. This can be achieved by utilizing complex variable theorem known as the argument pnnciple [75-80]. The Nyquist criterion follows directly from an important theorem of complex variable theorem, which states: If f(z) is analytic and dtfferent from zero on the contour C, then [75-80]
5.6
where N IS the number of zeros of f(z) lymg inside the chosen contour C and f (z) is the derivative of f(z). Furthermore, since f(z)
101
IS
analytic and further since
--------------------------------------------------------------------------
/(z) f(z)
=
d -- [lnf(z)]
...
dz
5.7
Eqn. 5.6 can be wntten as
f~&~ dz =
L'.Jlnf(z)]
=
j2rr.N
000
5.8
But L'.Jlnf(z )]
=
+ j L'.Jargf(z )]
L'.Jlnlf(z )I]
000
59
where L'.Jx] denotes the vanation in x around the contour C. Therefore, it can be readily be shown that
f~g~
dz
=
j L'.Jarg [f(z )]]
000
5.10
since C is closed contour, lnlf(z )I cannot change around it and therefore this term is zero [75-80]. The proof ofEqn. 5.10 together with further details about the theorem can be found elsewhere [75-801 . The argument principle can also be interpreted geometrically. Further details on this mterpretation are given elsewhere [76]. Consider now the z-transform of the sampled Impulse response of a lmear baseband channel, Y(z), given in Eqn 5.1. Y(z) is as defined earlier, a polynomial of g degree m z -I and can be expressed as a function of z-1,f(z- 1). Forthe application considered here, It IS necessary to identify the roots of f(z- 1) for which the magnitudes of z is greater than unity. The root-i~~UfJcauon algorithm suggested in Eqns. 5 6 and 5.10 is applied directly to [(z-1)'fg1ve the value of k (the number ofroots ofY(z) that lie outside the unit circle). The roots of[(z-1) whosemagnitudes are less than unity are, therefore, the roots of Y(z) which he outside the unit circle. The contour proposed here is taken to be the unit Circle m the z-plane.
102
Before applying the algonthm, Y(z) in Eqn. 5 1 is modified such that each z is replaced by e16 to give Y(S), where
...
5 11
...
5.12
Since
hence, Eqn. 5.11 becomes Y(6)= Yo + y 1(cos6-Jsm6) + =
Re(6)
. . + y,(cosge- 1 smg6)
+ lm(6)
513
where Re() and Im() are the real and the imaginary part ofY(S), respectively. From Eqn. 5.13, the argument of Y(S) can be calculated, such that;
=
arg[Y(S)]
The value
of~
~
=
tan-{~:~:n
...
5.14
in Eqn. 5.14 has an important function m this algonthm. It decides
how many roots he inside the umt circle (contour) when 8 vanes from 0- 21t Here, the number of roots equals the number of times the value of~ exceeds 21t. The Nyqmst cntenon descnbed above, has been Simulated by computer as follows: i-
Let m represent the number of roots Jymg inside the unit circle. This value initially has been set to zero together With the value of8 in Eqns. 5.11-5.13.
n-
The value of Y(S) as suggested in Eqn. 5.13 can be determined. Having determmed Re(S) and Im(S), the value Eqn. 5.14.
103
of~
can also be determmed using
m-
The value
of~
IS checked, and whenever
~
exceeds 27t, the value of m is
incremented by unity. IV-
The value of9 will be incremented by t-9, and steps1i and iii will be repeated, until 9 reaches 27t. If 9 = 27t, the algorithm will terminate and the value of m will be equal to k, where k represents the number of roots lying inside the unit circle.
Fig. 5.2 shows the flow diagram of the above procedure, which has been Simulated by computer program as shown in the appendices. The Nyquist cntenon has been tested by computer simulation over different models of telephone channels. The sampled impulse responses of these channels are g1ven in Tables 2.1 and 2.2. Throughout the tests the number of arithmetic operatiOns involved to decide how many roots he outside the unit circle have been calculated and recorded as shown in Tables 5.9-5.17. It is clear from these tables that the mam factors which affect the number of operatiOns aret.9 and the number of components in the sampled impulse response of the baseband channel (the value of g in Eqn 5.11 ). When the decision is correct, It has been found that the level of distortion in the channel has no effect on the number of arithmetic operations involved. However, channels mtroducing high level of d!stortion require a small value of t.9 and long impulse response to give the correct decision. Therefore, the results in Tables 5.95.17 represent all the channels tested, since all the channels give the correct decisiOn. It has been recogmzed that as the value of t.9 mcreases, the number of arithmeuc operatiOns mvolved w1ll decrease until .at a certam value of t.9 the algonthm fails to identify all the k roots inside the unit circle. Furthermore, reducmg the value of g has the same effect as mcreasing the value of t-9. Tests over channels 1-4 show that when the value of t.9 IS increased up to 20 degrees and g reduced to 11, the algonthm can still Identify the correct number of k roots for each of the channels tested. Throughout the tests the number of real arithmetic operations involved were calculated (addiUon, subtraction and multiplication), and recorded as shown in Tables 5.9-5.17. Furthermore, tests over channels 1-3, whose results are not recorded, have shown that when the value of g is reduced to 8 and t.9 mcreased up to 24 degrees the algorithm still can correctly Identify the exact number of roots.
104
Tests over channels 5, 6 and 8 have shown that the algorithm can correctly identify the roots with the maximum value for Ll9 set up to 14 degrees and minimum value for gas 19. The results of these tests which show the number of arithmetic operations required are exactly the same as those recorded in Table 5.13. Tests over channel 7 show that this channel reqmres more operations as compared with the other channels tested, since the minimum value of g and the maximum values of Ll9 are, respectively, 41 and 6 m order to identify the exact number of roots as shown in Table 5.14. The algorithm has also been tested over the minimum phase versions of channels 1-8 These channels should give a value ofkequal to zero. Therefore, the algonthm should give the same results over any minimum phase channel. It has been found that the variation of
as 9 increases from 0- 27t is entirely d1fferent from
non-m1mmum phase channels. In the latter channels vanes stead!ly as 9 increases whereas with the former channels
oscillate around zero. This phenomenon has
been considered m the termination of the algonthm before 9 approaches the value of 21t. The results of this algorithm over the minimum phase channels are as shown m Table 5.15-5.17, for d1fferent values of g. Fmally, the algonthm involves the determinatiOn of trigonometric equations (sin, cos, arctan); these functions for different values of 9 can be stored in a look-up table and can be called through the execution of the algonthm. Therefore, the results presented m Tables 5 9-5.17 have ignored the evaluation of these functions m the cost calculation. 5.3 ROOT-FINDING ALGORITHMS 5.3.1 Algorithm 1 Here, the general form of Newton's method (also called Newton-Raphson iteratiOn) is by far the most popular method in numencal analys1s for locatmg the roots (zeros) of polynomial equations [67]. This method approximates the roots through an iterative process. It IS extended to cope with complex-valued polynomial, f(z). Fig. 5.3 shows the geometric mterpretation of Newton's method for the simple case of a real root.
105
---------------------------------------------------------------------------------
The first step in th1s method
IS
to make the initial guess for the m'• root; this is
abbreviated as 'Ym,o· The lme tangent to f(z) at the pomt C'Ym,oJC'Ym,o))
IS
next
determined. The intersection of this line with the z axis is the point 'Ym,I· As seen by the geometric interpretation, 'Ym,I represents a better es!lmate of the root of f(z). By repeating this process, a much better estimate w11l be made. From Fig 5 3,
=
tanS
fC'Ym,o) 'Ym,! -Ym,o
where/C'Ym 0 ) denotes the derivative off(z) at z
=
'Ym,l
'Ym,O
= "fm
0•
-
. .
5.15
...
5.16
...
5.17
Hence
In terms of an iterauve algonthm,
'Ym,l + 1
=
'Ym,a
-
Apart from the geometnc denva!lon, Eqns. 5.15-5.17, may also be developed from the Taylor senes expansion. This alternallve derivation
IS
useful in that it provides
an insight into the rate of convergence of the method [65-70]. Let !:J.z be a suitable value such that
J("fm
0
+ !:J.z) = 0
...
5.18
Expandmg the above equatiOn by Taylor series gives
f("fm,O
+ .:'.z)
+ ...
106
5 19
where /(z) denotes the second derivative of f(z). When Ym,o becomes close to the value of the root the approximate version to the above equation is obtainable by truncating the series after the flrst denvative where the factors associated with !lz become very small. Furthermore, at the value of a rootf(Y,.,o + ru) =0. Therefore, From Eqn. 5.19, the value of !lz is giVen by -J(Y,.,o)
/(Y,.,o)
...
5.20
...
5 21
which can be solved for
'Ym,1+l
=
J(y,.,,)
'Ym,l
-
--
/(y,...)
which is identical to Eqn 5.17. The Iterative process defined in Eqn 5.21 is repeated unull Ym,•+t -y,.,.l < d, where d IS some small, real-valued quantity. When this is obtained, the process seems to have converged to a root of the polynomial, within the accuracy set by the value of d. An alternative stopping cri tenon has been also used to test if lf(y,...)l
>
f
.-
0
0 0
~
_,.
0 0
_.,
0
0 0
Legend
-
T
r
X
~·I
. r
fl,-1' fi,O' f l , l ' ' ' " ' f I ,g
1:
Fig. 5.40 Two-tap feedforward transversal filter
.
"
. ... .-. ... . '\ ... .... \ \ _,.
~
~
~
~
~
~
. ~
\
~
•:;- _,.
0
\
~
\ : ... I \ I _, \ _, .. f
. .. •: •.. ... . ... •. .: ... ... ... ~
_,.
Legend
~
'
Root 1
~Root 3
•
_,.
_,.
~
•
Number or llorollon (I)
. .
-..
.
• _,. ~
... ~
~
•. _, ~
..
. .; _,-··
~
~
. _,. •. _, . _., 0
0
~
• ...
~
Legend Root 2
_,.
... ...
\
1\ \\ I
I
I I
\
\
\
I
Legend Root 1
\ \
\
~~oiL
•Humber olltoraflon " (I)
..
Root 4
"
.
...
I
flg, 5 •.43 Performance of algorithm 5 with channe13
rtg. 5.41 Performance of algorithm 5 with channel1
..E •
('\
f
:
Root 1 _
Root
~
L
... ...
Root 4
~
\
.
•
.
Num!Jor of ll.rotlon (I)
"
Fig. 5 42 Performance of algorithm 5 with channel 2
''
-''
'' '' '' '' '' '' ''
'' ''
\
Legend Root 5
'
\ ~I
'' '' '' '' '' '' '' '' 1 ' " ollluollon (I) Humbor
\I
~Root
L
Root 1
B.2.2!LR~o!~--
___
8~!'Ls
Root 2
..
Fig. 5.44 Performance of algorithm 5 WLfh channe14
142
__ _ _ _ j
" _,
.. -··
~
Legend
.... .•
= •
a....J..L_.
Btlll.~t!_
Legend -11
llil..L_
!U1,!_ J!u.U_
Wll..-
!.•.!f.J L -
!ll!..!...._ !l.uJ_L_
!!!.!.!..!.!...
!.•.!U_-
!!.!1..!..!!.!!..ll...-
!!.!.1.]__
!tl''_., _.,
!~1!1_7_
ltuJ..!_
__
_.,
b.!!.L_,
Fig. 5.45 Pnformanct of algorithm 5 with channtl5
rig. 5.47 Perform a ne• of algorithm 5 with chonntl7
.
"
·-• . -·· : ,;
,;
•: .... :
-.
;;"'
••
-··
. -·· =
!.!.!!..!_.
!ll!...!..!!.!14_
ii -n
!.tlll_
•
a....J..L_. Root 11
;
. ...
!.•.!'-'- _
;:
!.!.!!..1L IW!..1L
~
f
;
!t.'l•-•---
•71
_.,
!.t!!..!..._
~ Root 10 _
';
!!.!!..!..._ !.•.!1_3_-
Legend
f
:. _,.
Legend
!\!.tll_
_.,
~ll..-
!.!..!!...L_
!t.!.I.J•••
ll.Ui.!.~
IWU..~--
_,.
_.,
Fig. 5.46 Performance of algorithmS with channel 6
Fig. 5.48 Perform one• of algorithm 5 with chann•l B
143
----------------------------------------------------------------------------------------
Operatton
Channell
Channel2
Channel3
Channe14
AddJ.uon&
393
393
317
483
560
548
468
738
Subtraction
Mult1pltcauon
TableS 1
Number of anlhmeuc operauons mvolved over channels 1-4 to g1ve the decu10n, when applymg
Schur algonthm
Table 52
Operat100
ChannelS
Channel6
Channel?
ChannelS
AddtUon& Sublracuon
3
193
3
3
MuluphcatJon
4
256
4
4
Number of anthmeuc operauons mvolved over channeb 5-8 to glVe the dectSIOO, when applymg
Schur algonthm
Table53
TableS 4
Table 55
Operat100
Channel I
Ol.anne12
Channel3
Channe14
AdditlOD & Subtracuon
463
357
245
127
Multtphcat!On
680
542
360
168
Number of anthmettc operauons mvolved over channels 1-4 to g1ve the declSlon, when applymg Schur algonthm With g set to 19
Operauon
Channel!
Channe12
Channe13
Channe14
Add1Uon& Subtract10n
343
97
97
97
Multipltcatton
500
128
128
128
Number of anthmettc operations mvolved over channels 1-4 to gtve the declS!On, when applymg Schur algonthm With g set to 14
Operauon
Channel I
Channel2
Channcl3
Channel4
Add111on & SubtractiOn
67
3
3
3
Mult1phcauon
88
4
4
4
Number of anthmel!C operaUons mvolved over channels 1-4 to gtve the dcctston, when applymg Schur algonthm w1th g set to 9.
144
Operatlon
Channel I
Channel2
ChaMel3
Channe14
Addtuon & Subtracuon
3
3
3
3
Muluphcauon
4
4
4
4
T•b1e 56
Number of anthmeuc operauons mvolved over channels 1·4 to gtve the dectston, when applymg Schur algonthm Wtlh g set to 5
Operatton
ChannelS
Channel6
Chatmel7
ChannelS
AddtUOn & Subtracuon
3
3
3
3
Mulupltcauon
4
4
4
4
Number of anthmettc operauons mvo]ved over channels 5·8 to gtve the dectston, when applymg
T•b1e 57
Schur algonthm Wtlh g set to the values between 4 and 25
Operatton Addttton& Subtracuon Multtpltcatton
Table 58
g=4 100
g=9 360
g=14 770
g=19 1330
g=24
g=29
1903
2900
128
468
1008
1748
2524
3850
Number of anthmeuc opcrauons mvolved over the mm unum phase verSion of channels 1·8 to gtve the dectston, when applymg Schur algonthm wtth dlfferent values for g
Operauon
4
Addtuon& SubtractiOn
5824
6 3904
8 3008
MulttphcattOn
5005 91
3355 61
2585
Dtv!Sion
Table 59
47 -
10
12
14
20
1989
1728
16 1536
18
2368
1349
1280
2035 37
1705
1485 27
1320 24
1155 21
liDO
31
20
Number of anthmettc operations mvolved over channels 1-4, for dUferent values of 6.9 Wtlh g set to 11
Operatton Addmon & Subtractton Multtphcatton Dtvtston
Table 510
4 6734
6 4514
8 3478
10
12
14
16
18
2738
2294
1998
1776
1554
20 1480
5915 91
3965 61
3055 47
2405 37
2015 31
1755 27
1560 24
1365 21
1300 20
Number of anthmcttc operattons mvolved over channels 1·4, for dlfferent values of 6.9 wtth g set to 13
145
----------------------------------------------------------------------------------------------------
Operauon
4
6
8
10
12
14
Addtuon & Subtracuon
7644
5124
3948
3108
2604
2268
16 2016
18 1764
20 1680
Muluphcauon
6825
4575
3525
2775
2325
2025
1800
1575
1500
DtVlSlOR
91
61
47
37
31
27
24
21
20
Table 5 11
Number of anthmeuc operations mvolved over channels l-4, for dtfferent values of ~9
v.1th g set to 15
Operatton Addmon &
4 8554
6 5734
8
10
12
14
16
18
20
4418
3474
2914
2538
2256
1974
1880
7735 91
5185 61
3995 47
3145 37
2635
2295 27
2040 24
1785 21
1700 20
SubtractiOn
Mult1phcatton DtVISlOO
T•ble 512
31
Number of anthmeuc operauons mvolved over channels 1-4, for different values of ~9 wtth g set to 17
Operattan
4
6
8
10
12
14
16
18
20
Addmon & SubtractlOn
9464
6344
4888
3848
3224
2808
2496
2184
2080
Multtphcauon
8645
5795
4465
3515
2945
2565
2280
1995
1900
Dtvtston
91
61
47
37
31
27
24
21
20
Table513
Number of anthmeuc operattons mvolved over channels 1-4, for dtfferent values of .19 wtth g set to 19
Operat10n
t.9=4,g=4L
69=4,g=39
69=6, g = 41
Add!tton & Subtractton
19474
18564
13054
Mult•plicatton
18655
17745
12505
DIVlSlOO
91
91
61
TableS 14
Number of anthmettc operations mvolved over channel?, for dtfferent values of .19 and g
146
Operation
4
8
12
16
20
24
Addttton & Subtractton
2211
1131
807
591
483
429
28 375
32 321
Mu1ttpltcatton
1845
945
675
495
405
360
315
270
Dtvtston
41
21
15
11
9
8
7
6
Table 515
Number of anthmettc operauons mvolved over the mm unum phase verSion of channels 1-8, for dtfferent values of .6.9 wtth g set to 9
4 3315
8 1735
12 1182
16 945
20
24
787
629
28 580
550
Mu1ttphcauon
2940
1540
1050
840
700
560
490
440
Dtvtston
42
22
15
12
10
8
7
7
Operatton Addttton& Subtracuon
Table 516
Number of anthmetlc operauons mvolved over the mm unum phase verSton of channels 1-8, for dtfferent values of .6.9 wtth g set to 14
Operation
4
8
12
Addttton& Subtractton
4365
2285
1557
16 1245
Multtplicatton
3990 42
2090 22
1425 15
1140 12
Dtvtston
Table 5 17
20
24
28
32
1037
829
725
725
950
760 8
665
665 7
10
7
Number of anthrnettc operattons mvolved over the mmunum phase verston of channels 1-8, for different values of .6.9 wtth g set to 19
Operatton
Channel!
Charmel2
Channel3
Channe14
Addtuon& Subtracuon
34109-
27607
39846
44618
Mu1upl.tcauon
29874
24160 448
34932
39122
566
626
Dtvtston
Table 5 18
32
586
Number of anthmeuc operattons mvolvcd when applymg algonthm 1 over channels 1-4, wtth d set to 1 x 10_.
147
---------------------------------- -----------
Root number
Channel!
Channel2
Channe13
Ch.annel4
Root 1
-131 0
-134 7
-168 9
-141 I
Root2
-127 6
-135 4
-1712
-141 2
Root3
-1346
-130 2
-142 5
-139 3
Root4
--
-140 7
-1461
-138 8
----
---------
-144 8
RootS Root6 Root7
~~--
RootS
----
---
-222 9 -1325 -1460
Roots accuracy obtarned usmg algonthm 1 over channels 1-4 wtth d set to 1 x 10.....
Table 519
Oper.mon
ChannelS
Channe16
Channel?
ChannelS
Addtuon& Subtracuon
64313
47536
68654
37066
Muluphcat10n
56506
41692
60244
32458
620
620
538
746
DtVlSIOD
Table 5 20
Number of anthmettc operauons mvolved when applymg algonthm 1 over channels 5-8, With d set to
Root number
tcr OJ.annel5
Charmel6
Charmcl7
Root 1
-92.40
-16193
-178 17
-97 40
Root2
-200 56
-65 85
-139 49
-113 83
Root3
-87 46
-67 97
-134 67
-173 04
Root4
-97 48
-154 56
-174 11
-133 68
RootS
-188 29
-7014
-165 35
-110 87
Root6
-94 97
-66 43
---
-120 05
Root?
-190 17
-74 80
-119 60
-102 63
RootS
-9674
-158 35
-114 43
-115 47
Root9
-8290
Root 12
------------
Root 13
--
Root 10 Root 11
TableS 21
1X
----------
ChannelS
-
-122 65
------
-115 01
Roots accuracy obtamed usmg algonthm 1 over channels 5-8 wtth d set to 1 x 10_.
148
-114 79
-171 90
-131 84 -121 75
OperatiOn
Channel I
Channe12
Channe13
Charmel4
Addtuon & SubtracUon
34973
28776
52890
46798
Muluphcauon
30632
25188
46410
41042
468
766
658
DtVlSlOR
Table522
584
Number of anthmeuc operauons mvolved when applymg algonthm 1 over channels l-4, wtth d set to 1 x 10""'
Root number
Channel!
Channel2
Channel3
Channe14
Root 1
-25077
-257 22
-32193
-23142
Root2
-238 20
-24161
-275 94
-228 54
Root3
-237 60
-244
so
-222 09
-221 81
Root4
-20000
-22263
RootS
------
-153 64
--
Root?
--
--------
---------
-220 89
Root6 RootS
Table 5 23
-222 90 -228 58 -230 62
Roots accuracy obtamed usmg algonthm 1 over channels 1-4 wtth d set to 1 x to""'
Operation
Channel 5
Channel6
Channel?
ChannelS
Addttton & Subtractton
67926
51301
59154
39935
Mult1phcauon
59692
45012
52394
34986
668
510
578
DtVISIOR
Table 5 24
792
Number of anthmeuc operations mvolved when applymg algonthm 1 over channels 5-8, Wtth d set to 1 x la""'
-
149
Root number
ChannelS
Channel6
Channel?
ChannelS
Root 1
-140 52
-313 04
-342 09
-156 58
Root2
-219 54
-212 73
-228 80
-210 53
Root3
-138 52
-215 06
-229 74
-338 49
Root4
-147 86
-294 71
-231 53
-249 30
RootS
-213 51
-217 22
-225 10
-208 21
Root6
-140 39
-213 50
----
-217 85
Root?
-356 94
-221 47
-223 98
-159 68
RootS
-143 43
-297 79
-22023
-214 87
Root9
-------
-229 56
---
-180 51
------
-309 48
-219 97
--
-176 69
-----
-24027
Root 10 Root 11 Root 12 Root 13
---
-222 68
Roots accuracy obtamed usmg algonlhm 1 over channels S-S With d set to 1 x 10~
TableS 25
Ope rat ton
Channel I
Channe12
Channc13
Ch.anne14
Addiuon &. Subtracuon
34565
28412
43731
46330
Mulliphcatton
30296
24890
38382
40656
DIVISion
536
422
562
604
Number of anlhmet1c operations mvolved when applymg a]gonlhm 1 over channels 1-4, w1th ~ set to 1 x lOo~;
Table 5 26
Root number
Channel I
Rootl
-131 0
Root2
-118 5
Root3
-121 2
-126 4
-91 9
-123 3
Root4
---
-130 5
-962
-114 2
-----· -----
-140 4
----
------
RootS Root6 Root7 RootS
TableS 27
-
--
Otannel2
ChaMe13
Cltannel4
-135 4
-92 2
-120 I
-124 5
-94 2
-126 0
Roots accuracy obtamed usmg algonthm 1 over channels 1-4 wtth ~set to 1x
150
-119 8 -1157 -109 8
tcr
Operatton
Channel5
Channel6
Channel7
ChannelS
AddJ.uon& SubtractiOn
67444
50726
57771
37127
Mulupl..!catwn
59298
44532
50658
32550
612
462
470
Dt.,.,ston
730
Number of anthmettc operauons mvolved when applymg algontltm 1 over channels S·S, wtth ~set to 1 x 10-4
TableS 2S
Root number
ChannelS
Channel6
Channel7
ChannelS
RootL
-lOO 24
-86 37
-9618
-79 82
Root2
-109 66
-103 38
-9259
-90 01
Root3
-112 95
-10167
-99 06
-87 82
Root4
-11441
-8267
-9456
-7629
RootS
-102 22
-9442
-90 33
-86 96
Root6
-106 84
-84 39
··-·
-75 13
Root7
-105 99
-91 00
-98 19
-83 55
RootS
-107 33
-87 48
-93 25
-83 61
Root9
----------
-9055
---
-8400
------
-97 28
-7664
···-
-84 83
-----
-82 85
Root 10 Root 11 Root 12
Rootl3
Table 5 29
-85 28
Roots accuracy obtamed usmg algonthm 1 over channels S-8 With~ set to 1 X I er'
Operatton
Channel I
Channel2
Channel3
Channel4
Addttion& Subtractton
35761
29920
53789
48439
Multtpl..!catton
31348
26216
47228
42514
448
732
634
DtVlSlOn
Table 5 30
-
556
Number of anthmettc operanons mvolved when applymg d.lgonthm 1 over channels 1-4, With ~ set to 1 x 10_..
151
Root number
Channel!
Root 1
-250 8
-2572
-168 9
-141 I
Root2
-238 2
-241 0
-1406
-141 I
Root3
-169 2
-2041
-144 6
-139 3
Root4
---
-2104
-146 0
-138 8
RootS
-
----
--
-144 8
-----
-2229
Root6
---
Root?
-----
RootS
Channel2
....
--
Channe13
Channel4
-146 0 -146 0
Roots accuracy obtamed usmg algonthm 1 over channels 1-4 wllh l; set to 1 x 10.....
Table 5 31
Operation
ChannelS
Channel6
Channel?
ChannelS
Addtuon& Subtractton
69796
51975
60832
40897
Multlphcallon
61372
45630
53360
35860
634
496
544
DtV!StOR
Table 5 32
760
Number of anthmettc operauons mvolved when applymg algonthm 1 over channels 5-8, wuh l; set to 1 x 10_.
Root number
ChannelS
Channel6
Channel?
ChannelS
Root 1
-164.56
-16193
-178 17
-134 88
Root2
-200 55
-14714
-171 82
-135 67
Root3
-169 4
-145 48
-16168
-173 04
Root4
-171 85
-154 56
-17411
-133 68
RootS
-188 29
-147 76
-165 35
-132 22
Root6
-168 83
-144 60
---
-128 55
Root?
-19017
-154 09
-17213
-132 89
RootS
-159 6
-158 35
-158 76
-137 22
Root9
-------------
-155 35
-129 05
------
---171 90
-126 80
----
-138 07
--
-131 84
----
----
-139 79
Root 10 Root 11 Root 12 Root 13
Table 5 33
-
Roots accuracy obtamed usmg algonthm 1 over channels 5-8 wtth l; set to 1 x 10_.
152
----------------------------------------------------------------------------------------------
Operation
Channel I
Channel2
O.anne13
Channel4
Addtuon&
40943
35147
59160
54043
35932
30840
51980
47472
488
772
676
Subtntcuon Mulupltcauon Dtvtston
596
Number of anthmettc operattons mvolved when applymg extended algonthm 1 over channels 1-4
TableS 34
Root nmnber
Channel!
Channel2
Channel3
Channel4
Root 1
-131 0
-253 65
-168 89
-30192
Root2
-248 66
-254 86
-208 45
-267 21
Root3
-245 95
-135 43
-272 85
-371 34
Root4
---------
-272 89
-316 26
-364 35
-----
--
-363 73
--
-2229
--
---
-244 92
RootS Root6 Root? RootS
----
-362 06
Roots accuracy obta.med usmg extended algonthrn 1 over channels 1-4
Table 5 35
Operattan
ChannelS
Otannel6
Channel?
ChannelS
Addtuon& Subrracuon
75371
58064
343837
46598
Multtphcauon
66302
51018
304080
40902
678
2008
590
DtV!SlOR
Tab-le 5 36
806
Number of anthmeuc operauons mvolved when applymg extended algonthm 1 over channels 5-8
Root number
ChatmelS
Channel6
Channel?
ChannelS
Root 1
-276 00
-16193
-178 17
-287 47
Root2
-365 91
-300 05
-259 67
-273 67
Root3
-312 80
-27719
-306 90
-173 04
Root4
-305 43
-320 30
-325 75
-248 0
RootS
-352 01
-336 82
-32714
-330 00
Root6
-171 06
-315 94
---
-216 16
Root?
-190 17
-278 31
-310 91
-32111
RootS
-268 85
-293 24
-309 78
-313.52
Root9
------
-263 94
-31167
-321 35
Root 10 Root ll
Root 12 Root 13
TableS 37
-
-----
----
-310 06
-262.23
--
--
-333 54
------
-----
-239 87
Roots accuracy ob-tamed usmg extended algonthm 1 over channels S-8
153
-295 30
Oper:lliOn
OlaMell
Channel2
OlaMel2
Olannel4
Addtuon& Subtracuon
3262
2125
3680
10923
Muluphcauon
2802
4428
3168
9494
DIVISIOn
118
172
128
270
Table 5 38
Number of anthmettc operauons mvolved when applymg algonthm 1 over truncated channels 1-4
Rootnwnber
Channel!
Channel2
ChaMel3
Channel4
Root 1
-1309
-156 9
·122 3
-132 3
Root2
-104 8
-121 0
-61 5
-144 1
Root3
-641
-156 9
-502
-103 6
Root4
·-·
-38 2
-264
-760
RootS
----
-··
-655
Root6
-
-·
Root7 RootS
·--·
----
----·--
-··
-440 -270 -240
Roots accuracy obtamed usmg a1gonthm 1 over truncated channels 1-4
Table 5 39
OperatiOn
ChannelS
Olannel6
Olatmel7
CbaMelS
Addttton & Subtracuon
20891
18195
19733
17316
Multtphcatton
18240
15868
17224
15110
352
372
310
DtvlSIOn
TableS 40
396
Number of anthmettc operauons mvolved when applymg algonthm 1 over truncated channels 5-8
Rootnwnber
ChannelS
OJ.annel6
OJ.annel7
Olannel8
Root 1
-8625
-161 92
-178 17
-13021
Root2
-7636
-141 80
-128 45
-142 36
Root3
-134 78
-14165
-92 52
-10014
Root4
-7743
-7973
-6523
-99 78
RootS
-6000
-8374
-6015
-9707
Root6
-48 97
-44 41
-30 94
-41 70
Root7
-2426
-25 12
-18
so
-6200
RootS
-2002
-16 7
-23 35
-74
Root9
--
-21 0
----
-22 68
Root 10
----
Root 11
-··
·-· ·-·
Rootl2
-· --
Root 13
Table 5 41
.
-··
-8 31
-
----
-30 00
·-·
--·
-65
Roots accuracy obtamed usmg a]gonthm 1 over truncated channels 5-8
154
-16 9
--------------------------------------------------------------------------------------------------------------,
Operation
ChaMell
Channel2
Channel3
Cbannel4
Add1Uon & Subtracuon
7352
12211
18038
19683
Muluphcatton
6320
10532
15602
17148
Dtvlston
140
216
292
222
Table 5 42
Number of anlhmeuc operations mvolved when applymg algonthm 2 over channels 1-4
Rootnwnber
Channel I
Cbannel2
Channel3
Channe13
Rootl
-131 0
-134 7
-168 9
-1411
Root2
-127 6
-135 4
-144 6
-141 2
Root3
-1206
-1302
-110 3
-1393
Root4
--------
-118 0
-121 0
-1372
---
-122 3
--
-------
-
--
-742
RootS Root6 Root? RootS
TableS 43
-
--
-94 0 -75 I
Roots accuracy obtamed usmg a1gonthm 2 over channels 1-4
Operat1on
ChannelS
Channel6
Channel?
ChannelS
Addmon &
60165
62402
75599
69485
51936
53562
65604
59636
988
1176
994
1350
SubtracUon
Mu1t1pbcauon Dlv!ston
Table 5 44
Number of anthmellc operauoos mvolved when applymg algonthm 2 over channels 5-8
Rootnwnber
ChannelS
Channe16
Channel?
ChannelS
Root 1
-188 94
-16193
-178 17
-11791
Root2
-200 56
-174 67
-139 49
-113 83
TableS 45
-
Root3
-189 62
-128 24
-134 67
-173 04
Root4
-187 80
-154 56
-174 11
-133 68
RootS
-188 29
-12750
-165 35
-110 88
Root6
-187 09
-125 ()()
-14617
-12005
Root?
-19017
-132 94
----
-111 53
RootS
-119 21
-158 35
-104 83
-115 47
Root9
-142 70
-12033
-121 34
Root 10
-----
---
-17190
-122 65
Root 11
--
----
----
-118 11
Rootl2
----
----
Root 13
----
----
Roots accuracy obtamed usmg algonthm 2 over channels S-8
155
------
-131 84 -121 75
Operatton
Channel!
Channel2
Channel3
Channel4
Addttton & Subtraction
5733
8783
11070
24977
Multtpltcat!on
4642
7132
8962
20252
DtvlSton
54
84
90
204
Number of anthmeuc operat.J.ons mvolved when applymg algonthm 3 over channels 1-4
TableS 46
Root number
Channel!
Channe12
Chatmel3
Chatmel4
Root 1
-131 0
-1119
-168 9
-108 I
Root2
-127 6
-135 4
-110 9
-107 7
Root3
-113 8
-107 3
-106 9
-1127
Root4
-1122
-lOO I
-103 8
---
Root7
----
RootS
---
------
---------
-131 I
Root6
-------
RootS
Table 5 47
--
-912 -668 -59 5
Roots accuracy obt:amed usmg algonthm 3 over channels 1-4
Operatton
ChannelS
Channel6
Channel?
ChannelS
AddttJOn & Subtraction
20912
17686
44677
23441
Multtpltcat!On
16882
14284
35986
19008
136
234
208
Dtv!Ston TableS 48
Kumberof anthmeuc operattons mvolved when applymg atgonthm 3 over channels 5-8
Root number
ChannelS
Channe16
Channel?
ChannelS
Root 1
-10434
Root2
-120 81
-161 93
-178 17
-138 59
-107 53
-10078
Root3
-114 40
-108 47
-106 49
-10158
Root4
-173 04
-112 93
-120 91
-102 96
-99 66
RootS
-106 82
-115 28
-96 68
-93 66
Root6
-91 00
-9626
---
-74 26
Root7
-47 97
----
-7268
-84 72
-53 35
-68 21
RootS
-
-
----
Root9
---
--
---
Root 10
--
-46 84
-39 34
Rootll
--------
------
---
Root 12 Root 13 Table 5 49
ISO
Roots accuracy obtamcd usutg a]gonlhm 3 over channels 5-8
156
-------
-5034 -3678
---
Operauon
Chatu1ell
Chatulel2
Channel3
Channel4
Addruon & Subtraction
12315
14650
29529
376:!6
Multtpltcauon
10566
12676
25830
32944
248
340
410
DtvtSlOn Table 5 50
260
Number of anthmeuc operauons mvolved when applymg algonthm 4 over channels 14
Root number
Channel1
Chatu1el2
ChaMel3
Channel4
Root 1
-131 00
-134 67
-168 89
-163 62
Root2
-127 61
-13543
-168 45
-51 38
Root3
-134 73
-13019
-168 99
-48 14
Root4
···-
-139 28
-172 01
-43 89
------
---
RootS
---
Root6
----
Root?
----
RootS
---
-46 01 -222 90
---
-45 24 -45 82
Roots accuracy obtamed usmg algonthm 4 over channels 1-4
Table 5 51
Operauon
Chatu1el5
Channel6
Chatu1el?
ChannelS
AddtllOn & Subtracuon
41245
50089
113452
54061
Mult1pltcauon
36206
43870
100086
47444
534
684
528
DrviSlon Table 5 52
380
Number of anthmeuc operauons mvolved when applymg algonthm 4 over channels 5-8
Rootnwnber
Chatu1elS
Channel6
Channel?
Root 1
-168 42
-16193
-17817
-9094
Root2
-187 66
-102 91
-147 99
-112 21 -173 04
Root3
-78 65
-10152
-68 33
Root4
-174 97
-144 49
-70 35
-87 62
RootS
-67 90
-9919
-128 22
-11116
Root6
-166 66
-9725
-73 64
-104 26
Root7
-190 17
-149 97
-160 88
-9145
RootS
-150 49
-172 62
-139 09
-8029
Root9
--------··----
-98 50
-138 06
-140 26
-··---
-189 15
-102 98
-179 90
-8630
----
-142 86
-80 89
----
---
-8513
Root 10 Root 11 Root 12 Root 13 Table 5 53
ChannelS
Roots accuracy obtamcd usmg algonthm 4 over channels 5-8
157
Operatton
Channel I
Channe12
Channel3
Channe14
Addmon& SubtracUon
4883
8181
10963
32478
Mulllphcauon
3978
6668
89!0
26386
68
78
226
DtVLSIOR
40
Number of anthmettc operauons mvolved when applymg algonlhm S over channels 1-4
TableS 54
Root number
Channel!
Root 1
-131 00
-102 61
-168 89
-104 99
Root2
-108 10
-135 43
-168 80
-105 43
Root3
-106 98
-9792
-17176
-130 16
Root4
-----
-98 53
-103 22
-131 73
RootS
··--
Root?
---
---------
-125 29
Root6
------
RootS
Channel2
Channel3
Channel4
-lOO 83 -!01 54 -100 29
Roots accuracy obtamed usmg algonlhm S over channels 1-4
Operation
ChannelS
Channel6
Channel?
ChannelS
Addtuon & Subtractton
43200
32574
70547
72911
Multiplicauon
35032
26436
57006
59234
2!0
318
512
DtvtSIOn
264
Number of anthmeuc operauons mvolved when applymg algonthm S over channels 5-8
Root number
ChannelS
Channe16
Channel?
ChannelS
Root 1
-11184
-16193
-178 17
-129 52
Root2
-12163
-11697
-123 92
-117 10
Root3
-116 21
-118 32
-179 02
-173 04
Root4
-113 30
-133 74
-114 10
-119 61
RootS
-110 09
-118 48
-167 52
-107 79
Root6
-108 88
-130 99
-!08 82
-122 25
Root?
-124 48
-135 09
-11077
-107 78
RootS
-110 lO
-!05 99
-115 55
-127 69
-108 68
-105 90
-11423
----------
-105 37
-115 68
Root9
--
RootlO
--
Root 11
---------
Root 12
Root 13
TableS 57
Roots accuracy obtamed usmg algonthm 5 over channels 5-8
158
-!05 89
-118 89
-10617
-102 91
---
-11035
-----------------------------------------------------------------------------
CHAPTER6
ADAPTIVE DECISION FEEDBACK EQUALIZERS
6.1 INTRODUCTION In the transmission of digltal data at the highest possible transmission rate over a bandlimited telephone channel, it is normally necessary to employ an equalizer at the receiver to correct the signal distortion introduced by the channel [3,27,32]. The equalizer is usually implemented in the form of lmear transversal filters and can contribute a considerable increase to the cost of the modem, especially when a microprocessor system is used in the implementation of the modem [33,84]. This chapter first discusses the adaptive estimation of the sampled impulse response of the linear baseband channel and then extends the discussion to include the adaptive adjustment of the decision feedback equalizer. Three different algorithms are presented and tested by computer simulation, for the adjustment of the tap gains of the decision feedback equalizer (Fig. 6.1). The first algorithm adJusts the tap gams of the equalizer such that, at the exact equalization of the channel, the signal-to-noise ratio at the detector input is maximum. The remaining algorithms attempt to minimize the mean-square error in the equalized signal by means of gradient and Kalman techniques.
6.2 FEED FORWARD TRANSVERSAL FILTER CHANNEL ESTIMATOR The first of the basic techniques for estimating the sampled impulse response of the channel uses an adaptive feedforward transversal filter [3,32-38] and is a development of the technique used by Magee and Proakis to estimate the complex vector, Y, (sampled Impulse response of the channel). It has (g+l) taps which IS equal to the number of components of the sampled impulse response of the channel, and the tap gains are adjusted in such a way as to minimize the mean-square error
159
-
between the received sampler, (Fig. 6.1) and its estimate. When perfectly adjusted, the tap gains are the components of the sampled impulse response of the equivalent, baseband discrete-time channel model. The adaptive feedforward transversal filter channel estimator is as shown in Fig. 6.2 and operates as follows [3,34-38]. Each square marked Tin Fig. 6.2 is a store that holds the corresponding detected data symbol {.f,_J for h = 0, 1, ... , g. Each time the stores are triggered, the stored values are shifted one place to the right At time t=iT, the estimator IS fed with the received sample r, and the detected data symbol .f,. The detected data symbols {.f.} are assumed to be correct, so that.f, =s, for each
i. Let f, _1 be the previous stored estimate of Y, then an estimate of r, at the output of the estimator can be formed by
f.
=
...
6.1
...
6.2
The error in f., which is
e,
=
r, -
f,
is scaled by a small positive quantity, 1:1, to give M,. The resulting signalis multiplied by each of the information symbols .f~-h at the appropriate tap (where "' denotes complex conjugate). The resulting products are then added to the corresponding components of the previous estimate, f, _1, to give the new stored estimate f., whose (h + I)" component is
... 6.3
for h = 0, 1, .... , g. The above equation is usually known as the stochastic gradient algorithm (which is derived from the steepest descent algorithm) [35,85] for adjusting the tap gains of the channel estimator. It minimizes the average mean-square error between f, and r,. The above algorithm itself is usually known as the least mean-square (LMS) algorithm.
160
The factor!!. in Eqn 6.4 is the step size or (the averaging parameter of the estimator) and it need not necessarily be a constant. It is usually a fairly small number (of order
!x 1o-3 ) and determines the amount of adjustment made to the tap gains. The smaller the value of!!., the smaller IS the effect of additive noise on Y, but the slower is the convergence of Y, towards Y [37]. In the form presented, it can be seen that the number of complex multiplications involved in the generation of Y, is (2g+3) per received sample. The estimation process descnbed above is probably the least complex of all possible methods, and involves relatively few operations per iteration [3,17,34-38]. Clearly, the feedforward transversal filter channel estimator can be implemented easily, and it is able to track slow variations in the channels response [86]. However, it is well known that when the input samples {s.} are highly correlated, the convergence of the estimator to the optimum value is slow [87]. In the steady-state operation, the rate of change of the tap gains will be small. As such, after the initial convergence of the estimator has been achieved, the processor which performs the estimation process may be allowed to be idle, the actual adjustment of the tap gains only being performed after the error signal exceeds some pre-determined limit or after a given number of symbols have been processed and detected [34-38]. In order to evaluate the performance of the estrmator, computer simulation tests have been carried out over models of eight telephone channels, where the objective is to determine the convergence rate of the channel estimator. The sampled impulse responses of the equivalent linear baseband channels are as shown in Tables 2.1 and 2 2. The tap gains of the channel esttm!!tOr are adjusted dunng a training period of 500 symbols at the start of transmission, during which there is no need to detect the data symbols. At the start of the training period the tap gains are all set to zero, this condition representing no prior knowledge of the channel. At the end of the training period the tap gains of the channel estimator are frozen and then held fixed over the following data signal. The constant!!. (Eqn. 6.3) is given the value of 0 004 for the first 240 symbols of the training signal, the value of 0 001 for the next 100 symbols and the value ofO 00053 for the remaining 160 symbols in tests over channels 1-4. However, tests over channels 5-8, have shown that a better performance for the channel estimator can be achieved when the constant!!. is given the value of 0.001 forthe first 160 symbols, the value of0.0005 forthe next 150 symbols and the value
161
----------
of 0 00015 for the remaining 190 symbols. These arrangements approach the best overall adjustment of the channel estimator for channels 1-8, over a traimng period of 500 symbols, with~ being permitted to take on three different values. The measure of performance of the channel estimator was the squared error between the known sampled impulse responses of the time invariant channel, Y, and the estimated sampled impulse response, Y., which ts defined here as,
=
... 6.4
Fig. 6.3 shows the convergence rate of the channel estimator when estimating the sampled impulse response of telephone channel 1 at signal-to-noise ratios of 15, 20, 25, 30, 35, 40, 45, 50, 55 and 60 dB. Tests with channels 2-8, have shown that the convergence rate of the channel estimator is approximately similar to that achieved over channel!, especially when the number of components in the channel estimator is the same. Therefore, the results of the convergence rate of the channel estimator over channel 1 have been considered to be representative of all the channels under test. Furthermore, the value of '\l1 (l=i) at the end of the iterative process has been calculated and recorded as shown in Tables 6.1 and 6.2, respectively for channels 1-4 and 5-8. The signal-to-ratio is denoted SNR, where
... 6.5
bearing in rrund that the mean-square values of s,, 2 (?,are 10 and 42, respectively for channels 1-4 and 5-8 and the mean-square of w, is 2 Jmtlally, decreases and then remains fixed, especially when there is an insufficient number of taps in the filter D (30 taps or less).
iii-
At low signal-to-noise ratios (30 dB or less), the number of taps m the filter D has no significant effect on the value of\jf. However, at high signal-to-noise ratios, and with channels that introduce severe distortion, the effect of
165
--------------------------------------------------------------------------
increasrng the number of taps m the filter D becomes quite Significant. This suggests that at high signal-to-noise ratios and with channels that introduce severe distortion, the decrease in the value of 'V is due both to the channel estimator and lmearpre-detection filter adjustment method (more taps in the filter D are required). It has been found from the tests that if there is a slight change in the estimate of the sampled impulse response of channel 4, the root-finding algorithm might fat! to locate all the zeros off (z) that lie outside the unit circle. This, of course, leads to a significant increase in the value of 'Jf. The problem has been identified and solved by using nine starting points instead of five starting points in the root-findingalgorithm. The latter starting points are used with channels 1-4 whereas the former starting points are those suggested for channels 5-8 (Section 5.3.5). Throughout the tests, the number of arithmetic operations mvolved in the adjustment of the linear pre-detection filter, that is, addition, subtracuon, multiplication and dtvis!On of real numbers were computed and recorded as shown in Tables 6.4-6.18. The number of arithmetic operations involved in the estimation of the sampled impulse response of the linear baseband channel has not been included in the latter Tables, since this number is equal to (2g+3) complex multiplications per iteration. As further tests, the tolerance of the equalizer to additive white Gaussian noise over the eight models of telephone channels were obtained as shown in Figs. 6.4-6.9. It is assumed here that the equalizer has 50 taps and that there are g taps in the linear filter D and F, respectively. By transmitting around 1.5 x 106 data symbols per curve in Figs. 6.4-6.9, and m some cases muc~ more, it has been possible to keep the 95% confidence limits for the different curves to less than about ±0.5 dB [88]. Extra tests have were also carried out, but now an infrnite number of taps in the filter D and perfect estimation of the sampled impulse response of the channel were assumed. Furthermore, the roots of the z-transform of the sampled impulse response of the channel have been calculated theoreucally [60] The equalizer, adjusted as just described, is referred to as the equalizer la. In the form suggested here, equalizer la is a nonlinear equalizer with tap gains set to those presented in Tables 3.1 and 3.2 (ZF equalizer) for channels 1-8. In tests with equalizer la, the tolerance of the equalizer to additive white Gaussian noise was obtained and plotted (Ftgs. 6.4-6.9).
166
Extra tests, whose results are not considered here, have also been earned out to determine the effect of the number of taps in the channel estimator (value of g) on the performance of equalizer 1. In tests over channels 1-3, a significant improvement in the values of 'l>I> 'JI and a fraction of a dB increases in the tolerance to additive white Gaussian noise, together with a remarkable saving in the number of arithmetic operations mvolved, can be achieved by reducing the value of g to 15. However, the improvement has not been obtained in tests with other channels (channels 4-8) in spite of the significant improvement in the value of u 1• The degradation in the performance of the latter channels, arising from reducing the value of g, is due to the fact that the z-transform of the sampled impulse response of those channels have more than eight roots lying outside the unit circle in the z-plane. These roots require more taps in the pre-detection filter to equalize, especially when their magnitudes close to unity. Furthermore, reducing the value of g will obviously degrade the roots location accuracy as seen from 5.3.1. Finally, in the tests with equalizers 1 and la over channels 7 and 8, it has been found that both equalizers suffer from very long error bursts. However, the results presented in Tables 6 3-6.18 show that the technique employed in equalizer 1 correctly adjusts the tap gains of the filters D and F. This suggests that the decision feedback equalizer can no longer be recommended as a detection process. It also suggests that a better tolerance for the system to additive while Gaussian noise can be achieved when the adjustment technique is combined with a more sophisticated detector, such as near maximum-hkelihood detector [17 ,31,51,55,59]. 6.3.2 Equalizer 2 This is based on the least mean-square stochastic gradient algorithm, which is extensively reported in the literature and is widely applied m practical communication equipment [3,25,27 ,32-33,89]. The adjustment ofthe tap gains of the linear feedforward transversal filters employed by the decision feedback equahzer is such that the mean-square error in the equahzed signal is minimum. This can be considered as the conventional practical approach towards the minimum mean-square error equahzer. This equalizer uses the error in the equalized signal 6.13
167
----------------------------------------------------------------------------------
to adjust the tap gains {dh} and {fi.} of the filters D and Fin Fig. 6.1. It is assumed here that the lmear filter Din the forward section is a linear feedforward transversal filter With (n+ 1)-taps and the linear filter Fin the feedback section is also a linear feedforward transversal filter with g taps. In the adaptive adjustment mode, and after the reception of r., the coefficients of the filters are adjusted recursively, in order to follow time variations m the channel response. This is done as follows. The error signal given in Eqn. 6.13 is first calculated. This error signal is then scaled by a factor (step size) 11,, and the resulting product is then used to adJUSt the coefficients of both fllters according to the steepest-decent algorithm [3,25,27 ,32] ;
for
h=O,l, . .
. ,n
...
6.14
and
J..,
for
h=l,2, . . . ,g
... 6.15
where dh.• and f. .• are the (h + 1)'h and the hrh tap gains, respectively, of the filter D and F and where* denotes compl,ex conjugate. The parameter 11, is a small posiuve constant which rmght be fixed or variable. The rate of convergence towards the desrred adjustment of the equalizer increases with 11,, but the accuracy of the adjustment decreases as 11, increases. Thus a suitable compromise between the two conflicting requirements must be accepted. The problem of interest here is whether or not it is, in fact, possible to achieve the reqmred setting that minimizes the mean-square error in the equalized signal by means of the gradient algorithm within a reasonable time. Computer simulation tests have been carried out to determine the performance of the equalizer when operating over the eight models of telephone channels. In all tests, the tolerance of the equalizer to additive white Gaussian noise was obtained. The tap gains of the equalizer are adjusted by means of a known random training signal of 10,000 and 13,400 symbols, respectively, for channels 1-4 and 5-8, at the start of the transmission, dunng which there is no need to detect the data symbols.
168
--
At the start of the training signal the tap gains are all set to zero, this condition representing no prior knowledge of the channel. At the end of the training signal the tap gams are frozen and held fixed over the duration of the data signal. The same signal-to-noise ratio is, of course maintained over the training signal as over the data signal. In tests over channels 1-4 the constant !'J., (Eqns. 6.14 and 6.15) is assigned the value 0.002 for the fust 3000 symbols of the training signal and the value 0.0002 for the remaining 7000 symbols. However, in tests over channels 5 and 6 the value of the constant is set to 0.0004 for the frrst 3000 symbols and 0.00005 for the remaining 10400 symbols. This arrangement approaches the best overall adjustment of the equalizer for channels 1-4 and 5-6, with a training signal of 10,000 and 13,400 symbols, respectively and with !'J., being permitted to take two different values. Tests with channel 7 have shown that it is possible to obtain a satisfactory adjustment of the equalizer only if the length of the training signal exceeds 25,000 symbols, but this number is far from acceptable in any practical system design. Furthermore, m tests over channel 8, it has not been found possible to achieve a satisfactory adjustment of the equalizer even when !'J., takes different values and the traimng signal exceeds 50,000 symbols. Ideally, under the assumptions made here, equalizer 2 can be adjusted as closely as required to minimize the mean-square error in the equalized signal, so long as the training signal is made sufficient long. In practice, however, telephone circuits are not strictly time invariant and, moreover, they introduce various transient effects, such as impulsive noise and occasional signal-level and carrier phase changes [5,27,90], which temporarily degrade the adjustment of the equalizer. The latter is also affected, to some degree, by errors in the detected data symbols [91]. Furthermore, even quite small drifts in the sampling phase of the received signal require a significant adjustment in the equalizer tap gains, thus setting an upper limit on the adjustment of the equalizer [92]. Since tests in which equalizer2 was adjusted during the detection of the data have led to inconclusive results, it was finally decided to adopt the procedure descnbed, in which about four seconds under reasonably ideal conditions are allowed for the adJUStment of the equalizer. The tolerances of the equalizer to white Gaussian noise have been obtained and plotted as shown in Figs. 6.4-6.9, respectively for channels 1-6. The number of taps
169
in filters D and F are the same as those used by equalizer 1. The results of equalizer 2 over channels 7 and 8, have not been presented here, smce it is not possible to achieve a satisfactory adJustment of the equahzerwithin four seconds trainingpenod.
6.3.3 Equalizer 3 A limitation of equalizer 2 is that it does not make full use of all the mformation available to it at the time of adaptation, making its rate of convergence relanvely slow [3,17,32-33]. More sophisticated techniques, such as theKalman filter can be used to overcome these limitations [3,17,32-33,93-94]. The Kalman algorithm is designed to update the tap gains of the linear feedforward filters employed by the equalizer, adaptively, from the received and previously detected signals at its input. The tap gains are adJusted such that the mean-square error in the equalized signal is a minimum [3,32,33,93-94].
r,_ 1
Letr,
r,_.and.f,
•••
§,_ 1
•••
.f,_ 8 betheinputtothefeedforwardand
the feedback sections of the equalizer, respectively, at t=iT. The input signals can be represented by the vector X,
= [
r,
r,_ 1
•
•
•
r,_.
s,
.f,_ 1
•
•
•
s,_ 8
]
•••
6.16
Also let C,_ 1 be an estimate of the tap gams of the equalizer at time t=(i-1)T
c,_l
= [
d,-1.o
d,-1.1
.
.
.
d,_l.•
f-1.1 f.-1.2 . . . f.-1.. l
... 617
The signal at the output of the equalizer at timet =(I-1)T is C, _1X,r, (where T denote the transpose), which may be different from the ideal output by an error e., where
...
6.18
It can be shown that the new estimate of the tap gams of the equalizer that mmim1zes the mean-square error in the equalized signal is
170
g~ven
by [3,32,93-94]
C,
=
C,_t
+
G,e,
...
6.19
•
...
6.20
...
6 21
where
• G,
P,_tx.r
=
ro+X,P,_tx;
and
P,
=
1 (J)
(P,_I
The symbol G, is a
-
G,X,P,_J
(n+g+l) · -component vector known as the Kalman gain and P, _1 is
the (n+g+ 1) x (n+g+ 1) covariance matrix of the error in the estimate of the actual equalizer coefficients. The parameter ro in Eqn. 6.21 and 6.22 is a positive constant less than unity, usually very close to unity, selected to provide short term averaging. Initially, the matrix P, is set to the identity matrix and C, is set to a null vector. Further details on the algonthm and on the derivation of Eqns. 6.19 and 6 21 are given elsewhere [3,17,32-33,93-94] As before, the performance of equalizer 3 was tested by computer simulation and the results of these tests appear m Figs. 6.4-6 9, respectively, for channels 1-6. In all tests, the tolerance of the equalizer to additive white Gaussian noise was obtained and plotted. It is assumed that the filter D and F has 50 and g taps, respectively. The value of the constant ro in Eqns. 6.20 and 6.21 was set to 0.9999. Furthermore, the tap gains are obtained by transmitting a known and random sequence of 10,000 symbols. After this training period, the tap gains are frozen and held fixed. Tests with channels 7 and 8 have shown that, although a correct adjusnnent of the equalizer setting has been achieved with a mean-square error ofless than -20 dB, the equalizer suffers from long error bursts. Therefore, the results showmg the tolerance of equalizer 3 over channels 7 and 8 have not been considered.
171
6.4 ASSESSMENT OF RESULTS AND DISCUSSION The results presented in Figs. 6.4-6.9 show the tolerance of the equalizers to additive white Gaussian noise. The relative merits of the equalizers can now compared in the light of these results, as follows; i-
The performance of equalizer 1 compared with that of equalizer la shows that there IS a degradation (a fraction of a dB) in the equalizer performance over channels 1-3. However this degradation increases slightly over channels 4-6. The degradation in the equalizer 1 over channels 1-3 is mamly due to the channel estimation error, u 1 whereas over channels 4-6 it is due to both channel estimation error and the fimte number of taps in the filter D. Therefore, a better performance over channels 1-3 can be achieved using a more sophisticated channel estimator whereas with channels 4-6 a better performance can only be achteved by using a better estimator together with a larger number of taps in the filter D. In the latter case, a larger number of taps are needed because channels 4-6 have more roots lying outside the unit circle than channels 1-3, and that some of these roots are very close to the unit circle, as shown in Tables 4.1-4.2.
ii-
At an error rate of 10-4, the performance of equalizer 2 is always inferior to that of equalizer 1. This degradation increases as the level of the distortion increases. However, at an error rate of 10-1, the performance of equalizer 1 becomes inferior, especially over channels introducing severe distortion (channels 4-6). Over typical channels, both equalizers achieve approximately the same performance.
m-
The performance of equalizer 1 over the range of the error rates tested in Figs. 6.4-6.9 is as good as that of equalizer 3 for channels 1-2 (the typical channels), but is inferior for channels 3-6 (severely distorted channels). However, the performance of equalizer 1 improves as the error rate decreases. Furthermore, for channels 1-6, equalizer 1 appears hkely to be at least as good as, or better than equalizer 3 at error rates of 10-6. The latter conclusion is derived from an extrapolation of the results presented in Ftgs. 6.4-6 9, since tests cannot be carried out at such low error rates.
172
iv-
The performance of equalizer 2 is always inferior to that of equalizer 3. The degradation in the performance of equalizer 2 over the error rates tested, is generally less than 1 dB for channels 1-3, but increases up to 5 dB at lower error rates for channels introducing more severe distortion (channels 4-6).
The tendency for the performance of equalizer 1 to improve, relative to that of equalizers 2 and 3, as the error rate in the{§,} decreases, can be explained as follows. At high error rates (> 10-2), the predominant factor that determines the error rate, for a given signal-to-noise ratio in the equalized signal, is the mean-square distortion in the latter signal. Thus, equalizers 2 and 3, which attempt to reduce this distortion, can be expected to operate well under these conditions. The latter effect
IS
very
significant with channels 4-8, where the mean-square distortion is very high. However, at low errorrates (< 10_.), the predominant factor that determines the error for a given signal-to-noise ratio in the equalized signal, is the peak distortion in the latter signal. In fact, the large maJority of all errors now occur with the sequences of data-symbol values that lead to the most severe distortion in one or more of the corresponding equalized signals. Although the mean-square error caused by equalizers 2 and 3 is small, the peak distortion can be quite large. Equalizer 1, on the other hand, (assuming the correct detection of the preceding data symbols which is normal under the assumed conditions), introduces no distortion. Thus the increased noise level in the equalized signal of equalizer 1 relative to the others is now more than offset by the relatively large distortion that is experienced from time to time in the equalized signal of equalizers 2 and 3. In the computer simulation tests over channels 1-3, it was found that the tap gains of equalizer 2 approximate quite closely to the corresponding taps of equalizer 3. This
IS
consistent with the good performance of equalizer 2 over those channels,
and suggests that for channels that do not introduce extremely severe distortion, the gradient algorithm is a cost effective technique for adjusting the equalizer because of its computational simplicity. However, over channels introducing severe distortion (such as channels 4-6) the performance of equalizer 2 is inferior to that of equalizer 3, between 2 to 5 dB. Furthermore, the gradient technique did not achieve a satisfactory adjustment of the equalizer tap gains over channels introducing more
173
---------- - - - - -
severe distortion, such as channels 7 and 8. Therefore, over channels introducing severe distortion, the gradient algorithm is no longer a useful technique for the adjustment of the equalizer. Equalizer 1 achieves a performance as good as that of achieved by equalizers 2 and 3 over typically distorted channels 1-3. However, over channels 4 and 6 the performance of equalizer 1, compared to that of equalizer 3, IS inferior by about 1 dB. This degradation in the performance mcreases as the distortion increases to about 2 dB in channelS. But equalizer 3 is considerably more complex than equalizers 1 and 2. However, vanous techniques are now available for simplifying the practical implementation of equalizer 3 and a substantial reducnon in complexity can be achieved by such methods [3,32,95-96]. Unfortunately, these techniques have a tendency towards a steady build-up in round-off errors, which can cause the system to become unstable [ 17]. Furthermore, even with such simphfications, the technique is still complex compared to that employed in equalizer 1. Therefore, it is clear from the above discussion that over channels introducing a typical amount of distortion, the technique employed in equalizer 1 can be considered for further development, whereas with channels introducing very severe distortion the decision feedback equalizer is no longer a suitable detection process. Fmally, a detailed study is now under investigation to develop the technique employed in equalizer 1 using the TMS320 family of DSP (Digital Signal Processor). Details on this study can be found elsewhere [97].
174
Linear baseband channel Y L,s,o(t-iT) Receiver filter
Transmission ath
{s,) Detector
~ to ..
Lioear filter D
Detected data symbols
{q,}
'
~
Lioear filter F
\..e r(t) ~-Sampler Received t = 1T
-
Linear filter adjustment method
.......
F
_t_y ~"'
Clhannelestimator
.
Fig. 6.1 Model of data transmission system
signal
A
T
1----__,._.
T
s,_2 ----------------·-~
A
Yl-1,0
Y1-1,1
C. e.
! - = . : c _ - - - f - - - - 1 - - - - - f - - - -.....--------------·---_.,
Fig. 6.2 Linear feedforward transversal filter channel estimator
Y1-1,g
0
-10
m
-20
"Cl
£
.,-
177
L.
-30
........ 0
c: 0
~
-40
0
E ;;
LJ
-50
-60
-70+-------.------.-------.------,-------.------.------~-------.------~----~ 0
50
100
150
200
250
300
350
Number of symbols
Fig. 6.3 Channel estimator performance over channel 1
400
450
500
~\
0,1
~~ \~
~\ ~~\ ~~\
0.01
-. CD
E
.,.... 0
~~\
0
.0
E
>-
\\~\
(I)
0,001
~\.
Legend
• • D
0
Equalizer 1 Equalizer~
Equalize.r~
0.0001 10
\\\.
;qualiZE!.!:.,2_
12
\\
14
16
18
20
Signal-to-noise ratio in dB
Fig. 6.4 Performance of equalizers with channel 1
178
22
0.1
'\~~
~
0 01
~\, ~~,'\ ~~\ ~\,\ ~\\
-".... .;
.... 0
.,........ 0
.0
E
>-
IJ'l
0.001
Legend
• •
Equalizer 1
D Equalizer 1a
0
0.0001 10
~ualiz~2Equalizer~
12
~
14
16
18
20
22
Signal-to-noise ratio in dB.
Fig. 6.5 Performance of equalizers with channel 2
179
,~, '~, \\
0.1
~
'\
0.01
\
-"' ....0
.... 0 ....
\
\\
\\\\, \'',\\
"'0
.n
E
>-
VJ
0.001
\\,\
Legend
• •
Equalizer 1
~
D Equalizer _!Q
0
~ualiz~2Equalizer~
\ \ \ ~
\ \ 0.0001 18
20
24
22
26
Signol-fo-noise rofio in dB
Fig. 6.6 Performance of equalizers with channel 3
180
28
0.1
~\
\
~
\\ \\ \
\
\
\ \ \
\ \ \
\ \ \ \\
0.01
-.
\
~
... e "' 0
.0
E (f)
I
\
\
I
0.001
Legend •
\
D Equalizer 1a
e
~ualiz~2-
0
Equalizer~
\ \
\
\
~·
I
Equalizer 1
\
\ I I
\
\ \
\
I
\I\I
\ \
\ \
0.0001+------,,-----,----t\lE,-I"I---r----.\--.... ---, 36 u 26 28 ~30 32 Signal-to-noise ratio in dB
Fig. 6.7 Performance of equalizers with channel 4
181
\ ~,
0.1
\
\
\
\
\
\
\
\
\ \ \ \ \ \ \
\ \ \\\ \ \\ \ \\ \ I\ \
0 01
-"......
I
(I)
...0 0"
\
I
..a
E
>-
(/)
\ \ \
I
0.001
Legend
•
Egualizer 1
~ualiz~2-
0 Egualizer l_
0.0001 26
28
\ \
I
0 Egualizer 1a
•
\
30
\ \
\
\ \
I
\ \ \
\
\
I
32
\ 34
36
38
Signal-to-noise ratio in dB
Fig. 6.8 Performance of equalizers with channel 5
182
40
-------·-----------------------------------------------------------------
0.1
0.01
-... .,;
~
...e " 0
\
.0
E
\
;;;-
\
\
0.001
\
Legend •
\
Equalizer 1
\
D Equalizer .!Q
e
~ualiz~2-
0
Egualize.r ~
\ \
\ \
26
28
30
32
34
36
38
Signal-to-noise ratio in dB.
Fig. 6.9 Performance of equalizers with channel 6
183
-
s:-...'R
Channel!
Channel2
Charmel3
Channel4
IS
-270
-270
-25 8
-253
20
-329
·32.9
-32.0
-319
25
-353
-35 4
-33 8
-33 7
30
-396
-397
-38 7
-38 I
35
-445
-446
-432
-431
40
-495
-497
-47 8
-46.7
45
-541
-54 2
-52.8
-53 4
so ss
-58 3
-574
-54 0
-581
-642
-637
-57 3
-53 6
60
-608
-62.3
-57 8
-561
Table 6 I
Error in !he estimate of !he sampl ed impulse response of channels 1-4.
51\'R
ChannelS
Channel6
Charmel7
ChannelS
IS
-242
-24 7
-21 3
-249
20
-29 6
-302
-266
-319
25
-344
-34 7
-31 2
-35 I
30
-37 s
-38 4
-35 8
-386
35
-42.1
-42.4
-40 I
-42.6
40
-477
-479
-39 4
-483
45
-52.8
-53 4
-456
-53 6
so ss
-503
-52.0
-406
-585
-58 0
-60 8
-42.7
-62 I
60
-593
-612
-449
-665
Table62
Error in !he esumate of !he sampl ed Impulse response of channels 5-8
-
184
Channel2
Channell
Channel3
Channel4
SNR
ljl,
ljl,
ljl
ljl,
ljl,
ljl
ljl,
ljl,
15
-32.1
-266
-25 s
-30 8
-27 s
-25 8
-28 3
-22.5
-21
20
-405
-319
-314
-38 6
-32.7
-317
-35 I
-28 4
ljl,
ljl,
ljl
s
-19 9
-17 3
-15 4
-27 s
-2M
-24 6
-22.4
ljl
25
-42.9
-344
-33 8
-42.4
-34 7
-341
-38 0
-30 I
-294
-300
-252
-23 9
30
-440
-394
-381
-42.1
-409
-38 s
-39 I
-35 9
-342
-29 6
-300
-26.8
35
-491
-442
-430
-508
-442
-344
-452
-397
-38 6
-32.2
-339
-300
40
-53 7
-491
-47 8
-537
-498
-483
-487
-446
-432
-31 8
-38 4
-309
45
-59 7
-52.9
-52.1
-58 4
-53 7
-52.5
-54 s
-492
-481
-29 8
-45 I
-29 6
so ss
~81
-551
-SS 4
~8
-551
-SS I
~16
-493
-491
-299
-477
-29 8
~7.7
-59 s
-58 9
~s
~0
-59 I
-519
-53 4
-52.1
-325
-442
-32.3
60
-71.8
-573
-571
~88
-58 8
-58 4
~10
-532
-52.5
-303
-466
-302
Table 6 3
Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and lmear ft.lter D m equahzer 1 usmg 30-t.aps m the lmear ft.lter D
Operat10n
Channel!
Channel2
Channel3
Channel4
Addtuon & Subtracnon
6908
10997
14027
38891 31504 366
Muluphcauon
5662
8968
11418
DtVISlOR
152
184
202
Table 6 4
Number of anthmettc opeanons mvolved m equallZer 1 wtth 30-taps m the lmear fllter D.
ChannelS
Channe16
ChannelS
Olannel7
SNR
ljl,
ljl,
ljl
ljl,
ljl,
ljl
ljl,
ljl,
ljl
ljl,
ljl,
ljl
15
-23 I
-216
-192
-23 0
-23 I
-20 I
-17 0
-11 8
-10 6
-10 9
-12.7
-87
20
-292
-263
-245
-260
-282
-240
-15 3
-16 s
-12.8
-170
-13 I
-14 s
25
-289
-306
-267
-209
-32.0
-205
-19 4
-19 8
-16 6
-15 7
-18 s -13 8
30
-25 I
-32.5
-244
-197
-353
-19 6
-22.2
-24 0
-200
-210
-20 8
35
-31 8
-382
-309
-19 9
-397
-19 9
-22.2
-29 I
-21 4
-8 8
-243
40
-24 8
-42.1
-24 7
-19 9
-449
-19 9
-22.7
-270
-21 3
-16 s -28 0
-16 2
45
-212
-47 s
-212
-19 3
-Sl 0
-19 3
-14 9
-287
-14 7
-10 5
-33 2
-105
-17 9 -8.7
50
-23 9
-451
-23 8
-19 2
-48 4
-19 2
-19 6
-283
-19 I
-13 3
-357
-13 3
55
-205
-52.6
-205
-19 3
-58 0
-19 3
-246
-292
-23 9
-13 4
-416
-13 4
60
-21 I
-53 7
-211
-19 3
-584
-19 3
-17 6
-285
-17 6
-46
-478
-46
• Table 6 S
Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and hnear fJ.lter D m equahzer 1 usmg 30-taps m the lmear ftlter D
Operatton
ChannelS
Channel6
Channel?
ChannelS
Addtuon & Subtractton
45683
40773
63353
76822
Multtpltcatton
37095
33097
51225
62425
Dwtslon
384
362
414
622
Table 6 6
Number of arithmetic opeanons mvolved m equalizer 1 wtth 30-taps m the lmear fllter D
185
Channel!
Chatmel2
Channel3
Channel4
SNR
v.
ljl,
'V
v.
v,
'V
v.
'V>
'V
v.
v,
'V
IS
·31 3
-25.7
-24 7
-300
-26 8
-25 I
-27 6
·22.5
·213
-20 I
·17 8
·IS 8
20
-42.5
-31 8
-314
·38 7
·32.7
-31 7
-343
·283
-273
-234
·23 2
·203
25
-41
s
·352
·342
-413
-35 6
·346
·372
-304
·296
·29 4
·25 4
-240
30
-443
·39 I
·38 0
-42.5
-405
·38 3
·391
-35 2
-337
·248
·295
-23
35
-485
-438
-42.6
·516
-437
-430
-460
·392
-38 4
·377
·33 7
·32.2
40
·551
-497
-487
.ss 8
-500
-490
-514
-445
-437
·28 4
-38 9
-280
45
-606
·53 4
·52.6
·59 6
-540
·53 0
·559
-492
-484
·307
-439
-305
50
-67 2
-562
·55 8
·64 6
·562
-556
-62.8
·500
-498
·302
-459
-30 I
55
-677
-59 I
·58 s
-681
·59 6
-59 0
·58 7
·561
·542
·291
-459
·290
60
-72.9
-58 I
-580
-68 8
·591
-587
-647
·53 3
-53 0
·293
-487
·29 3
Table 6 7
s
Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and lmear ftlter D m equahzer 1 usmg 40·taps m the lmear Illter D
Operat10n
Channel!
Channel2
Channel3
Addlt!On & SubtractiOn
7258
11001
14487
39234
Multtpltcauon
5942
8956
11778
31732
Dtvtston
172
200
222
382
Table 6 8
Channe14
Number d. anthmeuc opeanons mvolved m equalJ.Zer 1 wtth 40-taps m the hnear ftlter D
ChannelS
ChaMel7
Channel6
ChannelS
SNR
v.
ljl,
'V
v.
v,
'V
v.
'V>
'V
v.
ljl,
'V
IS
·24 9
·21 7
·200
-25 6
-226
-209
·18 I
·11 7
.lQ 8
·13 0
-8 7
·13
20
·29 5
-25 6
-242
-28 8
-274
·250
-204
·14 7
·13 6
·11 3
-17 s ·14 4
25
-33 3
·292
·27 8
·32.9
-30 5
-285
·22.5
-19 6
·17 8
-16 6
-17 3
·13 9
30
·366
-32.6
·312
-30 I
·352
·289
-23 6
·24 I
·20 8
-20 I
·19 7
·16.9
35
-39 3
-38 9
·36 I
·34 6
-406
·33 6
·23 7
-300
-22.8
·225
-24 6
-204
40
·360
-42.3
-351
-352
455
·349
-18 9
·28 7
·18 5
-206
·28 5
·19 9
45
·32.7
-476
·326
-369
·5Q5
·367
·209
-315
·205
·15 0
-34 6
·14 9
50
·31 0
-455
-309
-373
-48 3
-37 0
-171
-269
·16.7
-13 4
·377
·13 3
55
·362
·519
·361
-35 4
-56.1
·35 4
·31 4
-277
·26.1
·111
-41 8
-17 I
60
-307
-506
·306
-352
-58 3
·35 2
·166
-313
·16 8
·214
-48 9
-214
Table 6 9
Dtscrepancy between the actual and the esumated sampled tmpulse response of the channel and hnear ftlter D m equahzer 1 usmg 40-taps tn the lmear f'tlter D
Operatton
ChannelS
Channel6
Channel?
ChannelS
Addlllon & Subtractton
41493
37016
82478
73821
Multlphcatton
33631
29969
66576
59873
DtVlSlOil
372
512
610
Table 6 10
. 350
Number of anthmettc opeanons mvolved m equaliZer 1 Wllh 40-taps m the lmear fllter D.
186
Channel2
Chatu1ell SNR
IV·
IV>
Chatulel3
IV
IV·
IV,
IV
IV.
IV>
Channel4
IV
IV·
IV
IV·
15
-31 3
-246
-23 8
-302
-25 4
-242
-220
-266
-207
-203
-17 6
-15 8
20
-428
-313
-31 0
-387
-322
-31 3
-276
-34 2
-26.8
-27 0
-227
-213
25
-428
-35 6
-34 8
-428
-36.0
-351
-306
-372
-29 8
-299
-25 5
-242
30
-456
-39 9
-389
-437
-41 0
-39 2
-357
-407
-34 5
-32 3
-299
-279
35
-493
-454
-439
-513
-453
-443
-403
-459
-392
-403
-340
-33 I
40
-55 6
-485
-477
-54 5
-490
-47 9
-436
-53 9
-432
-35 8
-38 4
-33 9
45
-615
-52 9
-523
-595
-53 6
-526
-49 I
-56.4
-48 3
-411
-436
-39 2
50
-667
-566
-562
-64 8
-56 8
-562
-517
-655
-515
-406
-47 I
-397
55
-677
-584
-57 9
-715
-597
-594
-51 4
-607
-509
-412
-442
-39 4
60
-68 9
-58 5
-58 I
-671
-587
-58 I
-495
-593
-490
-41 0
-429
-38 8
Table 6 11
Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and lmear filter D m equahzer 1 usmg 50-taps m the lmear ftlter D
Operat10n
ChaMe11
Channel2
Channe13
Addtuon & Subtracuon
7608
11461
14946
42083
Multiphcatton
6222
9316
12138
33994
Dtvtston
192
220
242
416
Table 6 12
Channel4
Number of anthmeuc opeanons mvolved m equalLZer 1 wtth 50-taps m the lmearfllter D.
ChannelS
ChaMe17
Channel6
ChaMelS
SNR
IV,
IV· -21 6
-10 6
-10 I
IV· -13 0
IV
-202
IV· -19 6
IV,
-19 5
IV· -257
IV
-243
IV· -212
IV>
15
-10 8
-87
20
-307
-25 5
-243
-307
-27 5
-25 8
-229
-169
-15 9
-17 9
-17 6
-14 7
25
-336
-28 7
-27 5
-30 I
-30 I
-271
-23 0
-19 3
-17 8
-19 7
-16 6
-14 8
30
-365
-33 5
-31 8
-343
-36.1
-321
-254
-24 0
-217
-212
-221
-18 6
35
-31 9
-390
-312
-386
-407
-365
-221
-313
-216
-227
-259
-21 0
40
-318
-425
-314
-371
-46.1
-366
-19 3
-304
-18 9
-25 4
-28 8
-23 8
45
-35 6
-474
-353
-47 5
-4~9
-455
-19 4
-341
-19 3
-18 0
-35 2
-17 9
50
-33 5
-503
-33 4
-527
-528
-498
-24 3
-287
-23 0
-15 7
-357
-15 7
55
-36.7
-53 3
-366
-511
-54 4
-494
-19 5
-299
-19 I
-245
-41 I
-244
60
-367
-524
-366
-512
-569
-502
-18 7
-33 3
-18 5
-26 8
-51 I
-26 8
Table 6 13
IV
IV
Dtscrepancy between the actual and the estunated sampled tmpulse response of the channel and lmear filter D m equaltzer 1 usmg 50-taps m the lmear filter D.
Operatton
ChannelS
Channel6
Channel?
ChannelS
Addttton & Subtractton
46530
46073
83485
85284
Multtphcatton
37678
37290
67354
69125
DtVISIOQ
418
424
Table 6 14
534 -
7fJ2
Number of anthmettc opeanons mvolved m equaltzer 1 Wtth 50-taps m the lmear filter D
187
Channel!
Channel2
Channe13
Channel4
SNR
'I'I
'i'l
Ill
'I'I
'i'l
'I'
'Ill
'Ill
Ill
'Ill
'Ill
Ill
15
-319
-251
-24 3
-30 8
-26.6
-247
-271
-22.3
-21 0
-19 9
-17 6
-15 6
20
-402
-31 0
-305
-37 8
-317
-30 8
-352
-26.7
-26.1
-260
-22.4
-208
25
-449
-346
-342
-453
-34 8
-34 4
-37 8
-304
-297
-309
-25 4
-24 3
30
-443
-396
-38 3
-437
-403
-38 6
-409
-357
-34 6
-337
-306
-28 9
35
-507
-440
-432
-52.6
-441
-436
-46.3
-38 9
-38 2
-404
-32.8
-321
40
-563
-485
-478
-551
-492
-482
-53 4
-439
-434
-389
-38 I
-35 5
45
-603
-541
-53 2
-60 I
-544
-53 4
-55 6
-495
-486
-449
-430
-409
50
-650
-569
-563
-647
-571
-564
-669
-529
-52.7
-448
-46.2
-424
55
-689
-582
-57 9
-714
-59 8
-59 5
-603
-543
-53 3
-417
-453
-401
-691
-591
-587
-690
-604
-59 8
-53 8
-506
-489
-42.9
-432
-400
60
Table 6 IS
Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and hnear ftlter D m equabzer 1 usmg 60-taps m the lmear fllter D
Operatton
Channel I
Channel2
Cbannel3
Channe14
AddtUon & Subtractton
7958
12377
15674
47433
Mulupllcauon
6502
10048
12716
38290
Dtvtston
212
244
264
468
Table 6 16
Number of anthmeuc opeanons mvolved Ill equahzer 1 wuh 60-taps tn the hnear filter D
Cbannel S
Channel6
Channel?
ChannelS
SNR
'Ill
'i'l
Ill
'Ill
'Ill
Ill
'Ill
'Ill
Ill
'Ill
'Ill
Ill
15
-24 5
-204
-19 0
-25 5
-209
-19 6
-18 7
-Ill
-104
-12.9
-10 I
-8 3
20
-29 8
-263
-24 7
-302
-286
-263
-23 6
-16 5
-15 7
-16 8
-16 6
-13 7
25
-341
-28 8
-277
-33 6
-295
-28 0
-245
-19 6
-18 4
-19 9
-17 5
-15 5
30
-352
-34 5
-31 8
-37 I
-36.8
-340
-29 6
-257
-242
-215
-206
-18 0
35
-33 7
-403
-32 8
-426
-419
-392
-25 6
-29 I
-23 9
-19 3
-251
-18 3
40
-34 5
-42.1
-33 8
-432
-460
-413
-281
-24 7
-23 I
-22.3
-28 9
-215
45
-42.2
-476
-41 I
-50 8
-504
-476
-259
-33 4
-252
-22.4
-35 8
-22.2
50
-38 2
-484
-37 8
-55 5
-542
-51 8
-32.6
-30 I
-28 I
-15 3
-361
-15 3
55
-421
-50 9
-41 5
-55 8
-55 3
-52.5
-26.1
-28 6
-242
-16 8
-379
-16 8
60
-39 4
-54 2
-393
-602
-579
-559
-283
-33 2
-271
-303
-51 0
-303
Table 6 17
Dtscrepancy between the actual and the estunated sampled unpulse response of the channel and bnear filter D m equaliZer 1 usmg 60-taps m the lmear ftlter D
Ope rat ton
OtannelS
Channel6
Cbannel7
ChannelS
Addttlon & Subtractton
46157
38435
72709
86179
Multtpltcauon
37322
31003
58603
69751
Dtvtston
430
386
502
718
Table 6 lS
Number of anthmettc opeanons mvolved m equaliZer 1 With 60-taps tn the lmear ftlter D
188
CHAPTER 7 ADAPTIVE ADJUSTMENT OF THE DIGITAL RECEIVER WHEN OPERATING OVER HF RADIO LINKS 7.1 INTRODUCTION The radio frequency band in the region of 3·30 MHz is traditionally known as the High Frequency (HF) band and has for many years been used as a transmitting medium for long distance communication systems [3,5,7,17-24]. HF transmission is unpredictable due to the existence of multipath propagation and Rayleigh-fadmg [3,7]. However, the use of an HF channel as a transmission medmm is still of great importance, inspite of the introduction of transmission media such as satellite and optical fibre links, since HF links are both economical and flex1ble. In a high speed serial data transmission system operating over a 3 kHz band in the HF band spectrum, one of the major problems facing the modem designer is that of eliminating the intersymbol interference caused, by multipath propagation [3,7,98-99]. Therefore, when it is required to transmit data at the h1ghest poss1ble rate, it is necessary to employ an adaptive equalizer fortrackmg the sampled impulse response of the resulting time-varying baseband channel [3,17,22,98-99]. Traditionally, the tap gains of the equalizer have been adjusted adaptively using the gradient, Kalman or latice algonthms, to minimize the mean-square error in the equalized signal [3,17,32-33,98-99] . .Alternatively, an equalizer may be adjusted from the estimate ofthe sampled impulse response of the channel [17,27,39]. This chapter is pnmarily concerned with the latter type of equalizer. 7.2 MODEL OF SYSTEM Fig. 7.1 shows the model of a synchronous serial data transmission system which employs an HF radio link as the transmission path. The information to be transmitted is coded onto a set of data symbols {s,}
189
s, = s... + jsb,
."
7.1
...
7.2
where
S
a,,,
Sb,J
=
±1 -
or
±3
--J=T. The {s.}, is stansucally independent and equally likely to have any of their possible values. It is assumed that s, =0 for ig. The filter Fin the feedback section of the data receiver is a linear feedforward transversal filter whose tap gams at time instant t=(i+n)T are given by the g-component vector Ym+h,h = 0
for
193
. f. •• ..]
7.14
F, +• can be considered as an estimate of the sampled Impulse response of the channel
and filter D, which should be provided at the detector mput by the adaptive adjustment algorithm Ideally, when the filter D is correctly adjusted,
f. ••. h
e,.. n+h for
h = 1, 2, ... , g. With the correct detection of the data symbols and at the time instant t=(i+n)T, the output Signal from the filter F is an =
accurate estimate of the mtersymbol interference present in v, • ., such that the equalized signal is x, +•· Under these conditions
7.15
where q, •• is the output signal from the filter Fat t=(i+n)T. The detector itself then operates on the equalized signal which is: 7.16
to determine the detected data symbol
s,.
Three different fading channels have been studied and these are referred to as the HF channels 1-3. The signal over channel 1 is transmitted via three sky waves whereas the signal over channels 2 and 3 is transmitted via two sky waves. The frequency spread for channels 1 and 3 is 2 Hz, whereas for channel2 It is 1 Hz. The transrmssion delays
't 1 and 't2
of the sky waves relative to that of the shortest delay
sky wave, together with the other channels parameters, are as shown m Tables 7.1, where 0.4167 is the period Tin rmlh-seconds. Figs. 7.4-7 .6, show the characteristics of channels 1-3 over a duration of 25 seconds of transmissiOn.
7.3 ADJUSTMENT ALGORITHMS This section is concerned with the adaptive adjustment of the digital data receiver (filters D and F) shown in Fig. 7 .1. In Its ideal form the pre-detection filter is adjusted to be an all pass network, such that the resultant sampled impulse response of the
194
channel and filter is minimum phase [27 ,39]. Funhermore, the filter is such that in the resultant z-transform, all zeros of the z-transform of the sampled impulse response of the channel that lie outside the umt circle in the z-plane are replaced by the complex conjugate of their reciprocals, the remaining zeros bemg left unchanged [27 ,39]. Section 4.4 together with Section 5.3.5 descnbe a possible arrangement for the adjustment for the filters D and F shown in Fig. 7.1 when operating over time-invariant channels. However, when operating overtime-varying channels, such as those considered here, it is no longer possible, even with an infmlte number of taps, to achieve the ideal adjustment of the filter. In the tests here, it is assumed that the linear filter D has 50 taps whereas the filter F has g taps (g=29). It is also assumed that the channel estimator correctly estimates the sampled impulse response of the linear baseband channel such that;
Y,
=
Y,
7.17
Further details on the method for estnnatlon of the sampled impulse response of the HF link are given elsewhere [17 ,20,34,38,104]. Finally, tests have shown that the location and equalization of roots with absolute values greater than 1.05, rather than unity, degrades the performance of nonlinear equalizer by only a fraction of a dB [105]. However, it can be shown, that with this process, a remarkable saving in the number of arithmetic operations can be achieved, since the location of roots with absolute values less than 1.05 requires more iterations and a large number of taps are also required to equalize such roots. Therefore, 1t will be assumed throughout the following stydy that the adjustment algorithms equalize only those roots with absolute values greater than 1.05. Computer simulation tests have been carried out with the HF channels 1 and 2 using different algorithms for the adjustment of the filters D and F to achieve the objective described earlier. Three different sequences have been selected to represent the fadmg introduced by the given channels, to test the adjustment algonthms. These sequences are as follows, i-
Sequence 1 applies to channel 1. It has 400 values of Y, ranging from i=500 to i=900 (Fig. 7.4). Over the given sequence the value of I Y.l (measured in
195
dB) decreases to -7 dB and then increases again. The number of roots with absolute values greater than 1.05 increases from 5 to 7 and decreases to 5 again as shown in Fig. 7. 7. The exact number of roots has been calculated theoretically using the appropriate NAG library routine [60].
ii-
Sequence 2 is applies tochannel2. It has 400values off, ranging from i=3300 to i=3700 (Fig. 7.5). Over the sequence, the value of IY.l decreases to -12 dB and increases again. Six roots with absolute values greater than 1.05 appear simultaneously to g~ve a total of nine roots With absolute values greater than 1.05 during the sequence, as shown in Fig. 7.8.
iii-
Sequence 3 also is applies to channel2. It has 400 values of Y, ranging from i=5200 to i=5600 (Fig. 7 .5). During the sequence the number of roots with absolute values greater than 1.05 decreases rapidly from 9 to 3 as shown in Fig. 7.9.
7.3.1 Adjustment algorithm 1 The algorithm is that proposed in Section 6.3.1 with the same mne starting points as suggested for the telephone channels 5-8, to locate the roots of Y, (z) with absolute values greater than 1.05. It has been found that the algorithm with the given set of starting points always fails to locate all the required roots. Furthermore, the missing roots are usually very close to the unit circle but in certain cases the absolute values of the missing roots exceeds 1.25. The missing roots can have a serious effect on the performance of the adaptive receiver [105]. ThiS suggests that the nine starting points which have been employed for the telephone channels must be modified for the HF channel. In testing algorithm 5 (Section 5.3.5) over telephone and HF channels, it has been found that most of the roots of Y,(z) have being located using the first starting point (0.0
+ jO.O). Additional tests over HF channels have shown
that a better performance of the root-finding algorithm (that is, fewer missed roots), can be achieved when the last eight of the starting points move closer to the unit circle. The new set of nine starting points is as shown in Table 7.2. These points have been proposed by D. H. Lehmer [72], as centres of eight circles used to cover the unit circle, in checkmg whether or not there is a root inside the circle. However, tests have shown that a better performance can also be achieved by moving the eight
196
"
I ' !
starting points more closer to the unit circle than that suggested earlier in Table 7 .2. The latter set of starting points is referred to as set 2 whereas the former set is referred to as set 1. Tests over the HF channels 1 and 2 have shown that set 2 gives a slightly better performance than set 1, as can be seen from Figs. 7.7-7.9 and Tables 7.3-7.5. Tables 7.3-7.5, show the average number of arithmetic operations involved in the algorithm by the use of set 1 and set 2. 7.3.2 Adjustment algorithm 2 It has been found that, when operating over a time-varying channel, each of the roots at the time instant t=iT are very close to the corresponding roots at the time instant t=(i-1)T. Therefore, a considerable saving in the number of arithmetic operations involved in the adjustment algorithm can be achieved by using the roots at the time instant t=(i-1)T to locate the roots at time instant t=iT. This comes from the fact, supported by computer simulation tests, that these roots require very few iterations (1 to 3) for convergence to be achieved. However, there are roots appearing and disappearing from the region outside the unit circle. This suggests that the use of the previously located roots alone, for the given starting points from which the roots are located will not make any improvement to the performance of the algonthm. However, a novel arrangement have been proposed by S. F. Hau, which uses the previously located roots, has enabled the algorithm to give much improvement in its performance together with large saving in the number of arithmetic operations involved [106]. This arrangement can be described as follows; i-
In the first run of the algonthm (i=1), the value of~ is set to the frrst of the starting points in set 2 (Table 7 .2) and the algorithm starts looking for the roots as described in Section 5.3 .5. When the whole set of mne starting points has been used and the algonthm has not converged to a root, the algorithm will be terrrnnated and it will be assumed that all k roots are located. The values of the negative of the reciprocals of the located roots,{~m(i)}, will be stored in an array together with the nine starting pomts of set 2, as shown in Fig. 7.10.
u-
In the second run of the algorithm, (i=i+ 1), the value of~ is set now to the first value m the starting set given in Fig. 7.10, ~ 1 (1). This will converge very
197
quickly to its corresponding value,
p1(i + 1). If the roots corresponding to
P1(i + 1) are greater than 1 05 the value of P1(i + 1) will replace P1(i) in Fig. 7.10 otherwise the values of P1(i) will be discarded from the set given in Fig. 7.1 0. The next step is that the algorithm sets A. to ~(i) and this, of course, will converge to Its corresponding value of Pii + 1) and the process will be repeatedk times. After k such repetitions the algorithm will use the remaining nine starting points as described earlier. This of course first checks whether any more roots are left outside the required region and at the same time attempts to locate the given roots. If this check results in a root with modulus greater than 1.05, the negative of its reciprocal wiii be stored together with - the previous k{PmCi + 1)} in the starting array. The procedure just described is clanfied by the flow diagram given in Fig. 7 .11. Computer simulation test have been carried out with the sequences 1-3, using the algorithm 2 to adjust the tap gains of the filters D and F. The results of these tests show the number of roots tracked by the algorithm together with the average number of arithmetic operations involved in the algorithm and are given in Figs. 7.7-7.9 and Tables 7.3-7.5. The results clearly demonstrate the great saving in the number of arithmetic operations. The algorithm successfully tracks all the roots of sequence 3, whereas with sequences 1 and 2, it fails to locate all the required roots. However, the root-trackmg capability is much better than that of algorithm 1.
7.3.3 Adjustment algorithm 3 In the tests ofalgorithm2 with the sequences 1-3, it has been found that occasionally the algorithm fails to locate some of the roots of Y,(z) whose absolute values are greater than 1.05. Tests have shown that, at certam instants with the HF channels 1-3, the absolute values of the missing roots are as high as 1.16. As mentioned earlier, these missing roots can seriously degrade the performance if their absolute values become any larger. To improve the root-tracking capability of algorithm 2, a modified version to algorithm 2 is proposed. In the new arrangement the algorithm attempts to locate all the roots of Y, (z) that he outside the unit circle, using the same procedure as that descnbed in the previous section, but the roots with absolute values
198
greater than 1.05 are the only roots to be equalized. In other words, the latter roots are the only roots to be replaced by the complex conjugate of their reciprocals and to be used for the adjustment of the tap gains of the linear filter D. Algorithm 3 was tested in the same manner as algorithms 1 and 2 with the sequences 1-3, and the results of these tests appear in Figs. 7.7-7.9 and Tables 7.3-7.5. The results clearly show that algorithm 3 successfully locates most of the roots of Y,(z) with absolute values greater than 1.05, while algorithms 1 and 2 fail to locate some of these roots. The results presented in Figs. 7.7 and 7.9 show that algorithm 3 successfully locates most of the roots with absolute values greater than 1.05, when operating with the sequence 2. The results presented in Fig. 7.8 also show that, although the algorithm fails to locate all the required roots, the root-tracking capability is much better than that achieved by algorithms 1 and 2. However, the results presented in Tables 7.3-7.5 show a significant increase in the number of arithmetic operations involved in the algorithm compared with that required by algorithm 2. 7.3.4 Adjustment algorithm 4
Algorithm 4 is an improved version of algorithms 2 and 3. The approach considered here is to reduce the average number of arithmetic operations involved in algorithms 2 and 3. As mentioned earlier, algorithms 2 and 3 use the last nine starting points in the array shown in Fig. 7.10 to check whether any roots with absolute values greater than 1.05 appear or not and at the same time to locate these roots, if any. By observing the change in the number of r_oots With absolute values greater than 1.05, it has been found that most often the number is fixed for a very long period and sometimes this period exceeds 1000 samplmg instants especially over 2 sky waves HF channels (channels 2 and 3). Furthermore, it has been found that during such a period, the number of operations involved in using the remaining nine starting points in the array is the same as that involved when using the algonthm with the fust k startmgpoints in the array given in Table 7.10 to locate the roots. It should be borne in mind that the use of the remaining nine starting pomts here only gives a check that there are no more roots with absolute values greater than 1.05.
199
A useful reduction in the average number of arithmetic operations can be achieved through the use of the Schur algorithm described in Section 5.2.1, to check whether there is a root with a modulus greater than 1.05 [73]. This root-identification algorithm has been combmed With algorithms 2 and 3 as follows; i-
Initially, the new algorithm starts looking for the roots in the same way as before. The values of the k {~m(i)} are also stored in an array as shown in Fig. 7.10.
ii-
In the second run of the algorithm and after the whole set of k{~m(i)} will
be used to look for their corresponding values of k{~m(i + 1)}, the Schur algonthm is applied to F 1 (z ). If the Schur algorithm returns a decision that
F 1 (z) has no roots with absolute values greater than 1.05, the algorithm terminates without using the remaining set of nine starting points. However, if the Schur algorithm returns a decision that F 1 (z) has a root with a modulus greater than 1.05, the remaining set of nine starting points is used as for the algorithms 2 and 3. Computer simulation tests have also been carried out with the sequences 1-3 and algorithm 4. The results of these tests, which show the average number of arithmetic operations involved with the sequences 1-3, are given in Table 7.3-7.5. The results here assume that the Schur algorithm is combmed with algorithm 2. It has been found that when the Schur algorithm is combined with the algorithm 2, the same number of roots are tracked as by the algorithm 4. The same result also holds when the Schur algorithm is combined with the algonthm 3. Again, when using the Schur algorithm, the number of components in F 1 (z) is truncated to 10. This drop in the number of components, as shown in Section 5.2.1, has no effect on the decision reached by the Schur algonthm. It has also been found that algorithm 4 requires slightly more arithmetic operations than algorithm 2, especially when the Schur algonthm locates a root. However a great saving m the number of operations is achieved when the Schur algorithm decides that there are no more roots with absolute values greater than 1.05. This can be clearly observed when comparing the results presented in Tables 7.3-7.5. The selected sequences are of course those where roots cross the circle ofradius 1.05.
200
· 7.3.5 Adjustment algorithm 5 In testing of algorithm 2 with the sequences 1-3, it has been found that a useful reduction in the average number of arithmetic operations involved can be obtained by applying the adjustment algorithm every 4, 8, 12 and 16 sampling instants, as shown in Tables 7.6-7.8. The same saving, in the average number of arithmetic operations will, of course, also be achieved by the use of algorithms 3 and 4. The tests here also involve measurements of the error introduced in the sampled impulse response of the channel and the pre-detection filter by the given process. The measurement used are 'Jf2 and 'Jf3, where 'Jfz is as defined in Eqn. 6.11 and
7.18
The parameter 'Jf3 is a measure of the error introduced in
v,..
caused by the
components e, .. ,0, e,+., 1, • • • , e,+•.•-t• which should be all zero. The error here is normalized by taking it relative to the resultant of the magnitudes of
e..... , e, .. ,-+ 1, •• , e,+•.•+• [17]. The evaluation of 'Jfz and 'Jf3 for every sampling instant is complex and at the same time does not reveal much information about the performance of the algorithm, especially in a long test Therefore, the tests have considered only the worst values for 'Jf2 and 'Jf3 given by the algorithm. The results presented in Tables 7.9-7.11 demonstrate the effect on the values of 'Jfz and 'Jf3 , when algorithm 2 IS applied every 4, _8, 12 and 16 sampling instants, with the sequences 1-3. Similar effects are also observed when algorithm 3 or 4 is applied every 4, 8, 12 and 16 sampling instants. Although the sequences 2 and 3 have SIX roots crossing the circle of radius 1.05 simultaneously and sequence I has only two roots appearing and dtsappearmg, t}:le increase in the values of 'Jfz and 'Jf3 is much greater than with sequence 1. This is because sequence 1is given by a 2Hz frequency spread HF channel, whereas sequences 2 and 3 are given by a 1 Hz frequency spread HF channel From F1gs. 7.4-7.6, it can be seen that, with channels 1 and 3 (2Hz frequency spread) I Y,l changes more rapidly than with channel 2. It is clear from the results presented in Tables 7.9-7.11, that the effect of increasing the period, over which the algorithm is idle, has a greater influence on 'Jfz than those on 'Jf3•
201
A further useful reduction in the average number of anthmetic operations together with a significant decrease in the value of 'lf2 and 'lf3 compared with what presented earlier can be achieved through the application of prediction to the roots of Y,(z) that are being tracked by algonthms 2,3 or 4. Algorithm 5, based on such proposal, is a combination of degree-1 least square fading-memory pred1ct1on and algorithm 2 [17,20,38,104,107-108]. The algonthm locates and predicts the roots as follows; i-
Initially, let
Pm Cl)
and
P~(l) be the negative of the rec1procals of the mm
evaluated orpred1cted roots ofY,(z). Furthermore, let p~(O) = 0 form= 1, 2, ... , k.
il-
At the instant t = iT, apply algorithm 2 as descnbed m Section 7.3.2 to
detenmne the values of kfPm(l)} iu-
After the time mstantjT the degree-1least square fadmg-memory prediction is applied todetenmne the successive values of k{PmCi)}, k{P~(I + j, i)}. The latter can be ach1eved as follows; (1)
The error signal between the evaluated Pm(!) and its predicted value at time mstant t=(l-2J)T is calculated as ...
(2)
7.19
Following the calculation ofthe prediction error the one-step pred1ction of the rate of change of the PmCi) with respect to i is calculated using [17 ,20,38,104, 107-108]
...
7.20
where c1 is a posit1ve real constant in the range from 0 to 1, usually very close to 1 [17,20,38,104,105-106].
202
---- -------
(3)
Having calculated the rate of change of ~.. (i), the predicted value of~.. for the time instant t=(i+2j)T can be updated using ..
(4)
7.21
For the adjustment purposes it is necessary to know the values of k{~..} at time mstant t=(i+j)T. This can be achieved using
~~(i+j,i) = ~~(i+2j,i) iv-
-
~~~(i+2j,i)
...
7.22
At t=(i+j)T, the values of k{~~(i + j, i)} are used to adjust the tap gams of the linearfilterD and the values ofk{~~(i + j, 1)} form=!, 2, ... , k are stored in array as shown m Fig. 7.10.
v-
After time instant jT, algorithm 2, 3 or 4 is applied with the starting points returned from the previous step to determine the k values of {~.. (i + 2j)} and steps ii-iii are repeated.
Fig. 7.12 shows the logic flow diagram of algorithm 5 when the roots are evaluated and predicted every 4 sampling instants. Algorithm 5, has been tested by computer simulation and the results of these tests showing the worst values for"ljf2
and . "ljf3 over sequences 1-3 are shown in Tables
7.12-7.14. Throughout the tests the average number of arithmetic operations were calculated and these are recorded m Tables 7.15-7.17. It is assumed through the tests that degree-1least square fading-memory prediction is combined with the algorithm 2. Furthermore, the roots are evaluated every 16 samplmg instants, and are predicted every 16 sampling instants, but interleaved halfway between the measurements, to give an adjustment every 8 sampling instants. The constant c1 has the value of 0 92. The same improvement is gained when algorithm 3 or 4 is combined with the prediction algorithm.
203
It is clear from the results presented in Tables 7.12-7.17, that algonthm 5 saves a significant number of arithmetic operations and at the same time keeps the value of
'Jf2 as good as that achieved when algorithm 2 is applied. Furthermore, the effect on the value of 'Jf3 is as that when algorithm 2 is applied in those of prediction to give an adjustment every 8 sampling instants. The most difficult task in adJUStment using algorithm 5 is in the optimization of the value of c1, since the optimum value changes from channel to channel and from sequence to sequence.
7.4 DECISION FEEDBACK EQUALIZERS OPERATING OVER HF CHANNELS Here, the study presents different types of results which consider the effect of additive noise on the algorithms performance. The decision feedback equalizer discussed in Sections 3.4.1 and 6.3.1 was used here to investigate the performance of the algorithm. Several algorithms and techniques for the adjustment of the tap gains of the decision feedback equalizer were employed in the test. It is assumed that the estimate of the sampled impulse response of the channel and linear filter D has been employed by the hnear filter F. It is also assumed that the feedback filter has 29 taps. In order to obtain the most accurate comparison possible between the algorithms, the same fading sequence given by the values of the sampled impulse response of the linear baseband channel in Fig. 7.1 was used over the given number of 40,000 received samples {r,} that were employed in each test, but not, of course the same noise sequence. The equalizers under test were obtained by employing the ideal algorithm, algorithm 2 and algorithm 3 in the adjustment of the decision feedback equalizer. These equalizers are referred to here as HI, H2 and H3. In the equalizer HI, it IS assumed that the number of taps m the linear filter D is infmite whereas the number of taps in the feedback filteris g. It is also assumed that the all the roots of Y, (z) lying outside the unit circle are equalized. The latter roots were found theoretically using the appropriate NAG library routine [60]. In the equalizers H2 and H3, it is assumed
204
that algorithms 2 and 3 are used in the adjustment of the equalizers. It is also assumed in the equalizers H2 and H3 that the linear filter D has 50 taps and the roots to be equalized are only those with absolute values greater than 1.05. The results of computer simulation tests showing the relative tolerances of the equalizers to additive white Gaussian noise when operating over HF channels I and 2 are as shown in Figs. 7.13 and 7.14,respecuvely. The signal-to-noise ratio (SNR) in dB is given by
...7.23
bearing in mind that the mean-square value of s,
and
w, are 10 and 2cr'-,
respectively. Finally, the effect of applying the algorithm 2 every 8 or 16 sampling instants on the tolerance of the equalizer H2 to additive white Gaussian noise was also investigated and is as shown in Figs. 7.15 and 7 .16, respectively forthe HF channels I and 2. In Figs 7.15 and 7.16 the equalizers H2A and H2B, refer to equalizers where algorithm 2 is applied every 8 and 16 sampling instants, respectively. Furthermore, equalizer H5 refers to the equalizer obtained by using algorithm 5 in the adjustment of the tap gains of the equalizer. It is assumed in the latter equalizer that algorithm 5 evaluates or predicts the roots every 8 sampling instants. The most Important conclusions obtained from Figs. 7.13-7.16 can be summarized as follows; i-
The degradation in the performance of equalizer H2 and H3 over that ofH1, increases as the signal-to-noise increases. This degradation in the performance of the equalizers H2 and H3 is caused by the following factors, (1)
Equalizers H2 and H3 equalize only the roots ofY, (z) with absolute values greater than 1.05, whereas H1 equalizes all the roots ofY, (z) that lie outside the unit circle.
205
(2)
The limited number of taps in the linear filter D, which can causes some problems in equalizing the roots of Y,(z) that are oflower absolute values and when the number of roots of Y,(z) with absolute values greater than 1.05 is high.
(3)
Algorithm 2 or 3 in the equalizers H2 or H3, respectively occasionally failed to find all the roots off,(z) with absolute values greater than I 05.
(4)
The problem of selecting the best delay in Y., from Eqn. 7.12, for evaluating the tap gains of the fllters D and F. The above factors mdicate the reasons for the greater degradation in performance of equalizers H2 and H3 over channel 2, when compared with their performance over channel!. This degradation occurs because, with channel2, there are six roots with absolute values greater than 1.05 appearing and disappearing as shown on Figs. 7.8 and 7.9, during a relatively short period. To be equalized, these roots usually require more taps in fllter D, and at the same time, it is possible that the root-finding algorithm fails to locate some of these roots.
ii-
At high error rates, the dominant factor that produces errors in the detected data symbols is additive noise. Therefore, it is difficult to distinguish which algorithm is better at these error rates. However, at low error rates, the dominant factor is intersymbol interference. This can be greatly reduced by the use of better adjustment algorithms and more taps in the filter D.
iii-
From the results presented in Figs. 7.15 and 7.16, it is clear that, at very high error rates, it is possible to apply algorithm2, every 8 or 16 sampling instants without any fear of any undue performance degradation. However, at low error rates it is possible to apply algorithm 2 every 8 or 16 sampling instants, only with channel2, whereas with channel!, every 8 sampling mstants seems to be the !unit This suggests that the smaller the frequency spread, the longer is the period over which the adjustment algorithm can remain idle. Of course, with channel 2, the variatton of Y,(z) with i is much slower than that of channel!, as shown in Figs. 7.4-7.6.
206
1v-
The equalizer HS achieves, as expected, a performance which is better than that of equal1zer H2B and worse than that of equalizer H2A, as shown in Figs. 7.15-7 .16. At high error rates the performance of the equalizer HS is as good as that achieved by equalizers the H2 and H2A. However, at low error rates, the performance of the equalizer HS deteriorates when compared with that of the H2 and H2A equalizers. The deterioration is much more in channel 1 than in channel 2, due to the larger frequency spread.
7.5 ASSESSMENT OF THE RESULTS AND DISCUSSION i-
In tests of algorithm 2 with the sequences 1-3, it has been found that a considerable improvement in the value of \j/2 can be achieved through the use ofY•-•+A (Eqn. 7.4) in place ofY, (Eqn. 7.6), m the adjustment of the tap gains of the linear filters D and F at llme instant t=iT. The results of these tests also show that the improvement 1s dependent on the value of h. It has been found that the best value for h depends on the sequence (or the channel) under test, and it usually is in the range 3 to 6, as shown in Tables 7.18-7.20, respectively for sequences 1-3. It has also been found that the average number of arithmetic operations involved is the same as that when using the true Y, (Eqn. 7.6).
ii-
In tests of the equalizers H2 and H3 over channel 3, it has been found that incorrect operation of the adaptive filter occurs periodically. In parucular, there are now errors in the detected data symbols, even in the complete absence of noise, with a typical error rate of up to about 10-3 in the detected
-
data symbols. The errors occur during the deeper fades of the shortest duration, that is, wllh the most rapid fading rate. Different HF channels with the parameters sirmlar to that of channel3 also give the same results. During the deep fade in channel3, the following points have been observed; (1)
A sharp increase in the values of\j/2
and \j/3 , over a deep fade with a
peak error at the peak of the fade. However, at this instant, errors occur in the detected data symbols, even in the complete absence of the n01se.
207
(2)
During the period of the deep fade, the values of the four Gaussian noise wave forms(p 1(t),
pz(t),
p3(t)
and
p4 (t) Fig. 7.3) cross the zero
axis as shown in Fig. 7.17. (3)
The value of IY,l drops to -33 dB, which is such that, over a period of 50 samplmg intervals, the received signal level decreases by 12 dB to its lowest level and then increases agin by 12 dB.
(4)
As shown in Fig. 7.8, six roots with absolute values greater than 1.05 appear simultaneously during this deep fade, to give a total of nine roots with absolute values greater than 1.05. Tests with Y, --+h in place of that Y, with h havmg different values have shown that the error free cond!tion cannot now be achieved.
iii-
A further test with a different arrangement of algorithm 2 has been carried out. In this test, the value of the mean-square error in the equalized Signal was calculated and recorded as shown in Tables 7.21 and 7.22 for channels 1 and 2, respectively. The value of the mean-square error in the equalized signal is given by
000
7.24
The results clearly show the effect of applying algorithm 2 every 8 and 16 sampling instants on the value of
E.
It has been shown that there is an
improvement when algorithm 5 is evaluating or predicting the roots every 8 sampling instants compared with that of when algorithm 2 is evaluating the roots every 16 samplmg instants. Furthermore, the tests also show the effect of using Y,_n+h in place of Y,. It has been found that when his equal to 5, a lower mean-square error can be obtained, so that h=5 is assumed in Tables 7.21 and 7.22.
208
Linear baseband channel
,.............................................................................................................................................................................................................................................................................................................................. . .
:
:
I,s,o(t-iT)
:
' ~-,-.,..-+~
Transmitted data symbols
! !
Transmitter filter
Linear modulator
HF radio
link
Linear demodulator
Receiver filter
: :1
: 1-4-__.,
''' White Gaussian noise '' '' .............................................................................................................................................................................................................................................................................'' A
( s,) Detector
~
.
De tected data symbols
rB¥ (q,l
~
Linear filter
F
.... ~
Fi
-
( r,}
...." -
Linear filter D
tDi
Linear filter adJustment method
tvi Channel estimator
Fig. 7.1 Model of data transmission system
'
sampler t =tT
r(t) Recetved signal
•vu
80
~
m
~
60
" 0
., .,"
:;: ::J
-
40
0
213
-5
-"" ""0
·;:
m 0 ::::;: -10
-15
-20+-----------r-----------r---------~r---------~r----------.-----------,
0
10000
20000
30000
40000
Sampling Instant !IT!
Fig. 7.5 2 Sky waves Rayleigh fading HF channel 2
50000
60000
10
m ""0
.!: -10
-" >0
214
-c ""0 :I
-20
Ol
:::." -30
-40+-----------.-----------.----------,,----------,-----------,-----------, 60000 10000 20000 30000 50000 40000 0
Sampling Instant !IT!
Fig. 7.6 2 Sky waves Rayleigh fading HF channel 3
10
B
,,
0"
e .....0 ...
215
..r-
.a "
~
z
·-rj
6.
11 11
.--J
I I I
Legend
4
•::--------L
,,':!
•,
,,., ''•,
11
I
I
,;
1l
•:
I 11 :
:
I
I ' III : : ~ la 11 ~ I II : : 11 : ' Jl- - - - - ____ ,.__11- - - -11 ~ I I I ;~---.-.-1I
: :
:
.• J_,.. ____ _!::
I
t
..........!
I , ~I I I , 11 1 I: 11,111 I
~~.!!! !~
• ,
: :
u •:
--
• ............................. • -------------NAG roots
0 Set 1 roots
Set 2 roots
2
0 Algor[thm 2_roots t:. Algorithm 3 roots
0 0
100
200
Sampling Instant {lTI
Fig. 7.7 Variation of the number of located roots over sequence 1 for different algorithms
300
' 400
10
i j' i ·-,·-------;.----.----· : !r--·--·:r-·r:r:__ 1_J__________ . . ___ ,_________ ,
8
-I 11
'
" 15
e
216
r____!r'•f:':_ ___________f.!. __ .!.':... 1\
6
I
....0 .,...
_j
~--'-->
E
z"
.
4
L. •,.r----..l ............ .
J
,.......... ,......
:.:
II I
:
. .......
::..
::
11
1
,................,;
:
11~
:
l
_JI
.
I
.0
I I
::
-
.----------, L egend
•
...•
NAG roots
1 roots 0 Set.......................
e Set 2 roots ---------------
2·
0
Algor9thm
~
roots
!:::. Algorothm 3 roots
0~--~----~--~~~--~ 300 400 0
100
200
Sampling Instant
!JT!
Fig. 7.8 Variation of the number of located roots over sequence 2 for different algorithms
10
I. r"
I I
.-·-···· .......... •••• J ••• -.
B
.
1;
....e
6
:
,,
I'
I I
"-----,
..\
p---- - ················-·
I I I 1-1 I I I
............ ·t .....
"---:-,I I I
l-----------,
0
217
... "
..a
54
Legend
z
•
.
..................
NAG roots
I I I IL,_ I I I I
-
......
D Set-1 roots ...........
• -------------0 Set 2 roots
2
Algor\thm 2. roots
t:. Algorithm 3 roots 0 0
100
200
Sampling Instant !IT!
Fig. 7.9 Variation of the number of located roots over sequence 3 for different algorithms
300
400
r
The negative of the rcciprocals' r The fixed nine startmg points '
(
y
of the prevtously located roots
Value
~I
Order
I
(set 2) given m Table 7.2
I
k
2
k+l
2
9
k+2
k+9
Fig. 7.10 Starting points arrangement in algorithm 2
i= 0
J
•• I= 1+
1
ReadY1
No
Yes
i= I
; Use the nine startmg
Use the starung points
pomts set 2 gtven m
shown in 710 to locate the k ~m}
Table 7 2 to locate
F•r
~t' ~2'
•••••
~.
.
+ Dtscard the value 5f ~m or wluch I~ I< m I 05 and store the remaming
Store ~ , ~ , ••. , ~. in array togl;ther wttli the nme startmg pomts as shown m Ftg 7.10
ogether With the rune start.J.n
pomts as shown in Ftg. 7 10
+
+ Use the values ofk(~m} to adJUS! the tap gams of
hnear fi1tcrs D and F
+ Fig. 7.11 Flow diagram of algorithm 2 218
\
----------------------------------------------------------------------------
i=O,J=4 IC=3
+ i =I+ I
+ Ready;
+ IC=IC+ I NO
IC=S
NO
YES
IC=4
YES em(l) = J3m(1)- J3~(1,i-2j)
Apply algonthms 2, 3 or 4
+ -
•
2
J3m(•+2J,1) = J3m(l,i-2j) +(l-c1) em(~
+ J3~(1+2J,l) = J3~(1,1-2J) + J3~(i+2j,l) + (1-c~em(•)
Store the evaluated roots (J3m(•)) m array as shown m Fig. 7.10
+
. . I . J3m(l+J,i) = J3m(i+2J,1)- 2 J3m(i+2j,l)
+
IC=O
J
Use tl1e evaluated roots or tl1e pred1cted roots to adJust the lap gams of the filter D
+ Fig. 7.12 Flow diagram of algorithm 5
219
~ \
0.1
\
~
\\ \\
\\ \\ '\ '\
0.01
-"....
\\
I..
.... 0
\\ '\ '\
0 .a
E
>-
Vl
\\
0.001
1\
Legend
•
Equalizer H1
•
~uallz~H~
\
I \ I \
0 Equalizer H2
\\
0.0001
18
20
22
\
24
28
28
30
32
34
SNR in dB
Fig. 7.13 Performance of equalizers over HF channel 1
220
~
0.1
~
~
~
"""'"~ -
"~
"~
\\ \\
0.01
\~
\\
"....
~
0
....
\\ \\ \ \
....0
0"
.0
E
>-
Vl
\\
0.001
\
\
Legend
• •
Eguallzer Hl ~uallz~H~
0.0001 20
22
2'
\ \ \
\\
D Egualizer H2
18
\
I\ I \ 26
28
30
32
3'
38
SNR in dB
Fig. 7.14 Performance of equalizers over HF channel 2
221
---
-------------------------------------------------------
0.1
'\
'\
'\
'\ \
0.01
\
\\~\
\\
~\
-."......
~\ \
\\ ~\ \' \I \ \\\I \ \\\ I \ \\\ I
.... 0
0
.a
\
E
>Ill
0.001
Legend
\
•
Equalizer H2
•
fuualizE!!:. H28
\
0 Equalizer H2A 0 Equalizer HS
0.0001 18
20
22
2•
26
28
30
32
3
224
o.oo
a."' 0
-0.05
c--
.... --- --~-
E
Vl
-----Legend •
-0.10
- --t
P,(t)
D M!) __
• E...(IL 0 p,(t) -0.15-+--------...---------,--------.-----------, 200 0 50 100 150
Sampllng Instant !IT!
Fig. 7.17 Variation of pJt) with i
Channel
Number of sky waves
Relauve transmtsston delays of sky waves 1'1 and 'tz (ms)
Frequency spread
00,1.1,3 0
2
~----+-----------+-----~ 3
~--~2---4------~2-------4~--------0 0,3 0 3
2
I
00,30
2
Channels used m the tests
Table71
No
Set 1
Set2
1
0 0000 + J 0 0000
0 0000 + J 0 0000
2
08999 +)00000
09091 +JOOOOO
3
00000 - J 0 8999
00000 -)09091
4
00000 +j08999
00000 +)09091
5
-0 8999 + j 0 000
-0 9091 + J 0 0000
6
0.5890 - J 0 5890
06428 -j06428
7
0.5890 + J 0 5890
06428 +j06428
8
-0 5890 + J 0 5890
-0 6428 + J 0 6428
9
-0 5890 - J 0 5890
-0 6428 - J 0 6428
Table72
Startmg pomt sets 1 and 2 used m a1gonthm 1
Operauons
Algonthm 1 (set 1)
Algonthm 1 (set 2)
Algonthm2
Algonthm 3
Algonthm 4
Addtuon & Subtraction
20914
22163
13564
14564
10622
Multtphcauon
17202
18220
10800
11964
8535
Dtvtston
264
272
209
217
187
Table? 3
Average number of anthmeuc operations mvo1ved over sequence 1.
Algonthm 1 (set 2)
Algonthm 2
Algonthm3
Algonthm 4
1)
AddtUon & Subtraction
12187
12941
14790
18731
14974
Muluphcauon
12941
10710
12124
15339
11943
DtVlSIOR
217
222
214
234
209
Algonthm 2
Algonthm 3
Algonthm4
Operauons
Table 7 4
Algonthm 1 (set
Average nt.nTlber of anthmeuc operattons mvolved over sequence 2
Algonthm 1 (set
Algonthm 1 (set
1)
2)
Addttton & Subtracuon
23464
31743
15798
21998
12570
Multtphcatton
19279
26029
12937
17997
9969
DtvtSIOR
279
326
217
256
189
Operattons
Table 7 5
Average number of anthmettc operattons uwolved over sequence 3
225
- - - - - - - - - - - - - - - - - - - --
Operauon
EveryT
Every 4T
Every ST
Every 12T
Every 16f
Add!uon & Subtracuon
13147
3566
1818
1216
931
Mult!phcatton
10800
2927
1492
997
763
DtVISIOD
209
54
27
18
13
Table 7 6
Average nwnberof anthmeuc operations mvolved over sequence 1 when algomhm 2 applied every 1, 4, 8 or 16 samplmg mstants
Operatton
Every T
Every 4T
Every ST
Every 12T
Every 16T
Addmon& Subtracuon
14790
3615
1960
1353
1025
Mulupltcatton
12124
2966
1606
1109
840
DtvlSton
214
54
27
19
14
Table 7.7
Average number of anthmeuc operaUons mvolved over sequence 2 when algonthm 2 apphed every 1, 4, 8 or 16 samplmg mstants.
Operatton
EveryT
Every 4T
Every ST
Every 12T
Every 16T
Add!uon& Subtracuon
15798
3977
2155
1441
1110
Multlphcatton
12937
3256
1763
1179
908
DtviSton
217
54
28
18
14
Table7 8
Average number of anthmeuc operattons mvolved over sequence 3 when algonthm 2 apphed every 1, 4, 8 or 16 samplmg mstants
Parameter
Every T
Every4T
Every ST
Every 12T
Every 16T
"'' "''
-2850
-27 54
-2516
-2036
-17 04
-3250
-3123
-28 67
-26.38
-24 51
Table7 9
The worst values of 'Vz over sequence 1
and
'!f3 when afgonthm 2 apphed every 1, 4, 8 or 16 samplmg mstants
Parameter
Every T
Every 4T
Every ST
Every J2T
Every 16T
"''
-3140
-3138
-31 38
-31 38
-28 34
-31 82
-31 80
-31 60
-3160
-3125
"''
Table710
The worst values of 'lfz over sequence 2
and
'l'l when algonthm 2 applied every 1, 4, 8 or 16 samplmg mstanu
226
-
p arameter
Every T
Every 4T
Every 8T
Every 12T
Every 16f
'~'•
-3097
-30 11
-3011
-3015
·26.78
'If,
-3496
-2689
-26.37
-25 80
-2522
The worst values of 'Vz and '¥1 when algonthm 2 applied every 1. 4, 8 or 16 samphng mstants over sequence 3
Table 711
Parameter
Every 4T
Every ST
'~'•
-28 54
-2513
w,
-2929
-2529
The worn values of \jl2 and \jl3 when algonthm S evaluates or prechcts the roots every 4 or 8 samphng mstants over sequence 1
Table 712
Parameter
Every 4T
Every ST
'1'1
-3138
-31 38
w,
-3171
-31 43
Table'713
The worst values of 'l'z
and 'tf1 when algonthm S evaluates or predicts the roots every 4 or 8 samplmg
mstants over sequence 2.
Parameter
Every 4T
Every ST
'~'•
-3011
-30 11
w,
-26 89
-2637
Tablc714
The worst values of 'Vz and
'VJ when algonthm S evaluates or predtcts the roots every 4 or 8 samplmg
mstants over sequence 3
Table 715
.
Operation
Every4T
Every ST
Addttton& Subtractron
2611
1331
Multtphcauon
2148
1095
DtVlSIOO
48
24
Average number of anthmeuc operauons mvolved when algonlhm 5 evaluates or predtcts the roots every 4 or 8 samplmg mstants over sequence 1.
227
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - ----- --
Ope rattan
Every 4T
Every ST
Addmon& Subtracuon
2860
1442
Mu!uphcauon
2442
1185
D.vtston
48
24
Average number of anthmeuc operauons mvolved when algonthm S evaluates or predtcts the roor.s every 4 or 8 samplmg mstants over sequence 2
Table 7 16
Ope ratlon
Every4T
Every ST
Ad
